Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.5% → 94.4%

Time: 4.1s

Alternatives: 17

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (* (- y z) (/ (- t x) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y - z) * ((t - x) / (a - z)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Python

def code(x, y, z, t, a):
	return x + ((y - z) * ((t - x) / (a - z)))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y - z) * ((t - x) / (a - z)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 79.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x + \left(y - z\right) \cdot \frac{t - x}{a - z} \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (* (- y z) (/ (- t x) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + ((y - z) * ((t - x) / (a - z)))
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + ((y - z) * ((t - x) / (a - z)));
}

Python

def code(x, y, z, t, a):
	return x + ((y - z) * ((t - x) / (a - z)))

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + ((y - z) * ((t - x) / (a - z)));
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{z - a}\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, t - x, x\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\_1 \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z y) (- z a))) (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-229)
     (fma t_1 (- t x) x)
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) (+ x (* t_1 (- t x)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - y) / (z - a);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = fma(t_1, (t - x), x);
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = x + (t_1 * (t - x));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - y) / Float64(z - a))
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = fma(t_1, Float64(t - x), x);
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = Float64(x + Float64(t_1 * Float64(t - x)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], N[(t$95$1 * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(t$95$1 * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000007e-229

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      7. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. sub-negate N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      10. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      11. lift-*.f64 N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      12. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - y \cdot \frac{t - x}{z}\right) + t \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{t - x}{z} \cdot \left(a - y\right) + t \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a} - y, t\right) \]

      16. lower--.f64 52.7

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, a - \color{blue}{y}, t\right) \]

   6. Applied rewrites 52.7%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) \]

      4. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} \]

      8. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \cdot \left(t - x\right) \]

      9. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      10. frac-2neg N/A

          \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      11. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      14. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      15. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      16. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      17. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- z a)) (- t x) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-229)
     t_1
     (if (<= t_2 0.0) (fma (/ (- t x) z) (- a y) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / (z - a)), (t - x), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      7. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. sub-negate N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      10. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      11. lift-*.f64 N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      12. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - y \cdot \frac{t - x}{z}\right) + t \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{t - x}{z} \cdot \left(a - y\right) + t \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a} - y, t\right) \]

      16. lower--.f64 52.7

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, a - \color{blue}{y}, t\right) \]

   6. Applied rewrites 52.7%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 91.4% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(t - x\right)}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) (- z a)) (- y z) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -1e-229)
     t_1
     (if (<= t_2 0.0)
       (fma (/ (- t x) z) (- a y) t)
       (if (<= t_2 5e+306) t_1 (+ x (/ (* y (- t x)) (- a z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / (z - a)), (y - z), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -1e-229) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else if (t_2 <= 5e+306) {
		tmp = t_1;
	} else {
		tmp = x + ((y * (t - x)) / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -1e-229)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	elseif (t_2 <= 5e+306)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(Float64(y * Float64(t - x)) / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e-229], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$2, 5e+306], t$95$1, N[(x + N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999993e306

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. remove-double-neg N/A

         \[\leadsto \frac{t - x}{a - z} \cdot \left(y - z\right) + \color{blue}{x} \]

      8. lower-fma.f64 79.5

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)} \]

      9. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{t - x}{a - z}}, y - z, x\right) \]

      10. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(t - x\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, y - z, x\right) \]

      11. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(t - x\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, y - z, x\right) \]

      12. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(t - x\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      13. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      14. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{x - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      15. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, y - z, x\right) \]

      16. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{z - a}}, y - z, x\right) \]

      17. lower--.f64 79.5

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{z - a}}, y - z, x\right) \]

   3. Applied rewrites 79.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      7. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. sub-negate N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      10. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      11. lift-*.f64 N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      12. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - y \cdot \frac{t - x}{z}\right) + t \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{t - x}{z} \cdot \left(a - y\right) + t \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a} - y, t\right) \]

      16. lower--.f64 52.7

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, a - \color{blue}{y}, t\right) \]

   6. Applied rewrites 52.7%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

   if 4.99999999999999993e306 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{\color{blue}{a - z}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{\color{blue}{a} - z} \]

      3. lower--.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a - z} \]

      4. lower--.f64 54.4

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a - \color{blue}{z}} \]

   4. Applied rewrites 54.4%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 79.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ t_2 := \mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+246}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot \left(t - x\right)}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z)))))
        (t_2 (fma (/ (- y z) (- a z)) t x)))
   (if (<= t_1 -2e+246)
     (fma (- x t) (/ y z) t)
     (if (<= t_1 -1e-229)
       t_2
       (if (<= t_1 0.0)
         (fma (/ (- t x) z) (- a y) t)
         (if (<= t_1 5e+306) t_2 (+ x (/ (* y (- t x)) (- a z)))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double t_2 = fma(((y - z) / (a - z)), t, x);
	double tmp;
	if (t_1 <= -2e+246) {
		tmp = fma((x - t), (y / z), t);
	} else if (t_1 <= -1e-229) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else if (t_1 <= 5e+306) {
		tmp = t_2;
	} else {
		tmp = x + ((y * (t - x)) / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	t_2 = fma(Float64(Float64(y - z) / Float64(a - z)), t, x)
	tmp = 0.0
	if (t_1 <= -2e+246)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	elseif (t_1 <= -1e-229)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	elseif (t_1 <= 5e+306)
		tmp = t_2;
	else
		tmp = Float64(x + Float64(Float64(y * Float64(t - x)) / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+246], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$1, -1e-229], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$1, 5e+306], t$95$2, N[(x + N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2.00000000000000014e246

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 49.5

         \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{z}, t\right) \]

   9. Applied rewrites 49.5%

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]

   if -2.00000000000000014e246 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999993e306

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.7%

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t}{a - z}} + x \]

      5. mult-flip N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\left(t \cdot \frac{1}{a - z}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\left(\frac{1}{a - z} \cdot t\right)} + x \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot t} + x \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z}} \cdot t + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{y - z}}{a - z} \cdot t + x \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(z - y\right)\right)}}{a - z} \cdot t + x \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z - y\right)}\right)}{a - z} \cdot t + x \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\color{blue}{a - z}} \cdot t + x \]

      13. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(z - a\right)\right)}} \cdot t + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(z - a\right)}\right)} \cdot t + x \]

      15. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{z - y}{z - a}} \cdot t + x \]

      16. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{z - y}{z - a}} \cdot t + x \]

      17. lower-fma.f64 67.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t, x\right)} \]

   6. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)} \]

   if -1.00000000000000007e-229 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. mul-1-neg N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      5. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}\right)\right) + t \]

      7. div-sub N/A

         \[\leadsto \left(\mathsf{neg}\left(\left(\frac{y \cdot \left(t - x\right)}{z} - \frac{a \cdot \left(t - x\right)}{z}\right)\right)\right) + t \]

      8. sub-negate N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      9. lift-*.f64 N/A

         \[\leadsto \left(\frac{a \cdot \left(t - x\right)}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      10. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      11. lift-*.f64 N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - \frac{y \cdot \left(t - x\right)}{z}\right) + t \]

      12. associate-/l* N/A

          \[\leadsto \left(a \cdot \frac{t - x}{z} - y \cdot \frac{t - x}{z}\right) + t \]

      13. distribute-rgt-out-- N/A

          \[\leadsto \frac{t - x}{z} \cdot \left(a - y\right) + t \]

      14. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

      15. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a} - y, t\right) \]

      16. lower--.f64 52.7

          \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, a - \color{blue}{y}, t\right) \]

   6. Applied rewrites 52.7%

      \[\leadsto \mathsf{fma}\left(\frac{t - x}{z}, \color{blue}{a - y}, t\right) \]

   if 4.99999999999999993e306 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{\color{blue}{a - z}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{\color{blue}{a} - z} \]

      3. lower--.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a - z} \]

      4. lower--.f64 54.4

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a - \color{blue}{z}} \]

   4. Applied rewrites 54.4%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a - z}} \]
3. Recombined 4 regimes into one program.
4. Add Preprocessing

Alternative 5: 77.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a}\\ \mathbf{if}\;a \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) a)))))
   (if (<= a -5.7e-49)
     t_1
     (if (<= a 1.55e-7) (fma (- x t) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / a));
	double tmp;
	if (a <= -5.7e-49) {
		tmp = t_1;
	} else if (a <= 1.55e-7) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / a)))
	tmp = 0.0
	if (a <= -5.7e-49)
		tmp = t_1;
	elseif (a <= 1.55e-7)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -5.7e-49], t$95$1, If[LessEqual[a, 1.55e-7], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.7000000000000003e-49 or 1.55e-7 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{t - x}{\color{blue}{a}} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.1%

      \[\leadsto x + \left(y - z\right) \cdot \frac{t - x}{\color{blue}{a}} \]

   if -5.7000000000000003e-49 < a < 1.55e-7

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 77.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.45 \cdot 10^{-40}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{elif}\;a \leq 1500000:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -3.45e-40)
   (fma (/ (- y z) (- a z)) t x)
   (if (<= a 1500000.0)
     (fma (- x t) (/ (- y a) z) t)
     (+ x (* (- y z) (/ t (- a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -3.45e-40) {
		tmp = fma(((y - z) / (a - z)), t, x);
	} else if (a <= 1500000.0) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = x + ((y - z) * (t / (a - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -3.45e-40)
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), t, x);
	elseif (a <= 1500000.0)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -3.45e-40], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[a, 1500000.0], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.4499999999999998e-40

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.7%

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t}{a - z}} + x \]

      5. mult-flip N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\left(t \cdot \frac{1}{a - z}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\left(\frac{1}{a - z} \cdot t\right)} + x \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot t} + x \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z}} \cdot t + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{y - z}}{a - z} \cdot t + x \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(z - y\right)\right)}}{a - z} \cdot t + x \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z - y\right)}\right)}{a - z} \cdot t + x \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\color{blue}{a - z}} \cdot t + x \]

      13. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(z - a\right)\right)}} \cdot t + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(z - a\right)}\right)} \cdot t + x \]

      15. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{z - y}{z - a}} \cdot t + x \]

      16. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{z - y}{z - a}} \cdot t + x \]

      17. lower-fma.f64 67.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t, x\right)} \]

   6. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)} \]

   if -3.4499999999999998e-40 < a < 1.5e6

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   if 1.5e6 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.7%

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 75.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)\\ \mathbf{if}\;a \leq -3.45 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1500000:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) t x)))
   (if (<= a -3.45e-40)
     t_1
     (if (<= a 1500000.0) (fma (- x t) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (a - z)), t, x);
	double tmp;
	if (a <= -3.45e-40) {
		tmp = t_1;
	} else if (a <= 1500000.0) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(a - z)), t, x)
	tmp = 0.0
	if (a <= -3.45e-40)
		tmp = t_1;
	elseif (a <= 1500000.0)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[a, -3.45e-40], t$95$1, If[LessEqual[a, 1500000.0], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.4499999999999998e-40 or 1.5e6 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.7%

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z} + x} \]

      3. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t}{a - z}} + x \]

      4. lift-/.f64 N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\frac{t}{a - z}} + x \]

      5. mult-flip N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\left(t \cdot \frac{1}{a - z}\right)} + x \]

      6. *-commutative N/A

         \[\leadsto \left(y - z\right) \cdot \color{blue}{\left(\frac{1}{a - z} \cdot t\right)} + x \]

      7. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right) \cdot t} + x \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\frac{y - z}{a - z}} \cdot t + x \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\color{blue}{y - z}}{a - z} \cdot t + x \]

      10. sub-negate-rev N/A

          \[\leadsto \frac{\color{blue}{\mathsf{neg}\left(\left(z - y\right)\right)}}{a - z} \cdot t + x \]

      11. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\color{blue}{\left(z - y\right)}\right)}{a - z} \cdot t + x \]

      12. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\color{blue}{a - z}} \cdot t + x \]

      13. sub-negate-rev N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\color{blue}{\mathsf{neg}\left(\left(z - a\right)\right)}} \cdot t + x \]

      14. lift--.f64 N/A

          \[\leadsto \frac{\mathsf{neg}\left(\left(z - y\right)\right)}{\mathsf{neg}\left(\color{blue}{\left(z - a\right)}\right)} \cdot t + x \]

      15. frac-2neg N/A

          \[\leadsto \color{blue}{\frac{z - y}{z - a}} \cdot t + x \]

      16. lift-/.f64 N/A

          \[\leadsto \color{blue}{\frac{z - y}{z - a}} \cdot t + x \]

      17. lower-fma.f64 67.1

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t, x\right)} \]

   6. Applied rewrites 67.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{a - z}, t, x\right)} \]

   if -3.4499999999999998e-40 < a < 1.5e6

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.7 \cdot 10^{-49}:\\ \;\;\;\;x - \left(x - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.7e-49)
   (- x (* (- x t) (/ y a)))
   (if (<= a 9.6e+23)
     (fma (- x t) (/ (- y a) z) t)
     (+ x (* (- y z) (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.7e-49) {
		tmp = x - ((x - t) * (y / a));
	} else if (a <= 9.6e+23) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = x + ((y - z) * (t / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.7e-49)
		tmp = Float64(x - Float64(Float64(x - t) * Float64(y / a)));
	elseif (a <= 9.6e+23)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.7e-49], N[(x - N[(N[(x - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.6e+23], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.7000000000000003e-49

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t - x}, x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      4. sub-negate-rev N/A

         \[\leadsto x + \frac{y}{a} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x - t\right)\right)\right)} \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{a} \cdot \left(x - t\right)\right)\right)} \]

      6. sub-flip-reverse N/A

         \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(x - t\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(x - t\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto x - \frac{y}{a} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right)} \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{y}{a} \cdot \left(t - x\right)\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right) \cdot \frac{y}{a}}\right)\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{a}} \]

      12. sub-negate-rev N/A

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

      13. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(x - t\right) \cdot \frac{y}{a}} \]

      14. lower--.f64 48.3

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{x - \left(x - t\right) \cdot \frac{y}{a}} \]

   if -5.7000000000000003e-49 < a < 9.6e23

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   if 9.6e23 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.7%

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{t}{\color{blue}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.3%

      \[\leadsto x + \left(y - z\right) \cdot \frac{t}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-49}:\\ \;\;\;\;x - \left(x - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.5e-49)
   (- x (* (- x t) (/ y a)))
   (if (<= a 9.6e+23) (fma (- x t) (/ y z) t) (+ x (* (- y z) (/ t a))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.5e-49) {
		tmp = x - ((x - t) * (y / a));
	} else if (a <= 9.6e+23) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = x + ((y - z) * (t / a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.5e-49)
		tmp = Float64(x - Float64(Float64(x - t) * Float64(y / a)));
	elseif (a <= 9.6e+23)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.5e-49], N[(x - N[(N[(x - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.6e+23], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.50000000000000031e-49

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t - x}, x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      4. sub-negate-rev N/A

         \[\leadsto x + \frac{y}{a} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x - t\right)\right)\right)} \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{a} \cdot \left(x - t\right)\right)\right)} \]

      6. sub-flip-reverse N/A

         \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(x - t\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(x - t\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto x - \frac{y}{a} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right)} \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{y}{a} \cdot \left(t - x\right)\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right) \cdot \frac{y}{a}}\right)\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{a}} \]

      12. sub-negate-rev N/A

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

      13. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(x - t\right) \cdot \frac{y}{a}} \]

      14. lower--.f64 48.3

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{x - \left(x - t\right) \cdot \frac{y}{a}} \]

   if -5.50000000000000031e-49 < a < 9.6e23

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 49.5

         \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{z}, t\right) \]

   9. Applied rewrites 49.5%

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]

   if 9.6e23 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in x around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]
   3. Step-by-step derivation

   4. Applied rewrites 63.7%

      \[\leadsto x + \left(y - z\right) \cdot \frac{\color{blue}{t}}{a - z} \]

   5. Taylor expanded in z around 0

      \[\leadsto x + \left(y - z\right) \cdot \frac{t}{\color{blue}{a}} \]
   6. Step-by-step derivation

   7. Applied rewrites 43.3%

      \[\leadsto x + \left(y - z\right) \cdot \frac{t}{\color{blue}{a}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 70.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{-49}:\\ \;\;\;\;x - \left(x - t\right) \cdot \frac{y}{a}\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -5.5e-49)
   (- x (* (- x t) (/ y a)))
   (if (<= a 1.5e-7) (fma (- x t) (/ y z) t) (fma (/ y a) (- t x) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -5.5e-49) {
		tmp = x - ((x - t) * (y / a));
	} else if (a <= 1.5e-7) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = fma((y / a), (t - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -5.5e-49)
		tmp = Float64(x - Float64(Float64(x - t) * Float64(y / a)));
	elseif (a <= 1.5e-7)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = fma(Float64(y / a), Float64(t - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -5.5e-49], N[(x - N[(N[(x - t), $MachinePrecision] * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.5e-7], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -5.50000000000000031e-49

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   7. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{y}{a}, \color{blue}{t - x}, x\right) \]

      2. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{a} \cdot \left(t - x\right) + x} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{x + \frac{y}{a} \cdot \left(t - x\right)} \]

      4. sub-negate-rev N/A

         \[\leadsto x + \frac{y}{a} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(x - t\right)\right)\right)} \]

      5. distribute-rgt-neg-out N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{a} \cdot \left(x - t\right)\right)\right)} \]

      6. sub-flip-reverse N/A

         \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(x - t\right)} \]

      7. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{a} \cdot \left(x - t\right)} \]

      8. sub-negate-rev N/A

         \[\leadsto x - \frac{y}{a} \cdot \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right)} \]

      9. distribute-rgt-neg-out N/A

         \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\frac{y}{a} \cdot \left(t - x\right)\right)\right)} \]

      10. *-commutative N/A

          \[\leadsto x - \left(\mathsf{neg}\left(\color{blue}{\left(t - x\right) \cdot \frac{y}{a}}\right)\right) \]

      11. distribute-lft-neg-out N/A

          \[\leadsto x - \color{blue}{\left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \frac{y}{a}} \]

      12. sub-negate-rev N/A

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

      13. lower-*.f64 N/A

          \[\leadsto x - \color{blue}{\left(x - t\right) \cdot \frac{y}{a}} \]

      14. lower--.f64 48.3

          \[\leadsto x - \color{blue}{\left(x - t\right)} \cdot \frac{y}{a} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{x - \left(x - t\right) \cdot \frac{y}{a}} \]

   if -5.50000000000000031e-49 < a < 1.4999999999999999e-7

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 49.5

         \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{z}, t\right) \]

   9. Applied rewrites 49.5%

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]

   if 1.4999999999999999e-7 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 69.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{-49}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.5 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (- t x) x)))
   (if (<= a -5.5e-49) t_1 (if (<= a 1.5e-7) (fma (- x t) (/ y z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), (t - x), x);
	double tmp;
	if (a <= -5.5e-49) {
		tmp = t_1;
	} else if (a <= 1.5e-7) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -5.5e-49)
		tmp = t_1;
	elseif (a <= 1.5e-7)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.5e-49], t$95$1, If[LessEqual[a, 1.5e-7], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -5.50000000000000031e-49 or 1.4999999999999999e-7 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]
   5. Step-by-step derivation

      1. lower-/.f64 48.3

         \[\leadsto \mathsf{fma}\left(\frac{y}{\color{blue}{a}}, t - x, x\right) \]

   6. Applied rewrites 48.3%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{a}}, t - x, x\right) \]

   if -5.50000000000000031e-49 < a < 1.4999999999999999e-7

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 49.5

         \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{z}, t\right) \]

   9. Applied rewrites 49.5%

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 63.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;a \leq -6 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y t) a))))
   (if (<= a -6e-40) t_1 (if (<= a 9.6e+36) (fma (- x t) (/ y z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * t) / a);
	double tmp;
	if (a <= -6e-40) {
		tmp = t_1;
	} else if (a <= 9.6e+36) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y * t) / a))
	tmp = 0.0
	if (a <= -6e-40)
		tmp = t_1;
	elseif (a <= 9.6e+36)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -6e-40], t$95$1, If[LessEqual[a, 9.6e+36], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.00000000000000039e-40 or 9.5999999999999997e36 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) \]

      4. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} \]

      8. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \cdot \left(t - x\right) \]

      9. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      10. frac-2neg N/A

          \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      11. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      14. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      15. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      16. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      17. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{\color{blue}{a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a} \]

      3. lower--.f64 44.1

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a} \]

   6. Applied rewrites 44.1%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a}} \]

   7. Taylor expanded in x around 0

      \[\leadsto x + \frac{y \cdot t}{a} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.8%

      \[\leadsto x + \frac{y \cdot t}{a} \]

   if -6.00000000000000039e-40 < a < 9.5999999999999997e36

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   7. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   8. Step-by-step derivation

      1. lower-/.f64 49.5

         \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{z}, t\right) \]

   9. Applied rewrites 49.5%

      \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 49.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y \cdot t}{a}\\ \mathbf{if}\;a \leq -1.06 \cdot 10^{-40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{-12}:\\ \;\;\;\;\frac{y \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* y t) a))))
   (if (<= a -1.06e-40) t_1 (if (<= a 1.2e-12) (/ (* y (- x t)) z) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * t) / a);
	double tmp;
	if (a <= -1.06e-40) {
		tmp = t_1;
	} else if (a <= 1.2e-12) {
		tmp = (y * (x - t)) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + ((y * t) / a)
    if (a <= (-1.06d-40)) then
        tmp = t_1
    else if (a <= 1.2d-12) then
        tmp = (y * (x - t)) / z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y * t) / a);
	double tmp;
	if (a <= -1.06e-40) {
		tmp = t_1;
	} else if (a <= 1.2e-12) {
		tmp = (y * (x - t)) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y * t) / a)
	tmp = 0
	if a <= -1.06e-40:
		tmp = t_1
	elif a <= 1.2e-12:
		tmp = (y * (x - t)) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y * t) / a))
	tmp = 0.0
	if (a <= -1.06e-40)
		tmp = t_1;
	elseif (a <= 1.2e-12)
		tmp = Float64(Float64(y * Float64(x - t)) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y * t) / a);
	tmp = 0.0;
	if (a <= -1.06e-40)
		tmp = t_1;
	elseif (a <= 1.2e-12)
		tmp = (y * (x - t)) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y * t), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -1.06e-40], t$95$1, If[LessEqual[a, 1.2e-12], N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.06e-40 or 1.19999999999999994e-12 < a

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. *-commutative N/A

         \[\leadsto x + \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} \]

      3. lift-/.f64 N/A

         \[\leadsto x + \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) \]

      4. mult-flip N/A

         \[\leadsto x + \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) \]

      5. associate-*l* N/A

         \[\leadsto x + \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} \]

      6. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} \]

      7. lower-*.f64 N/A

         \[\leadsto x + \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} \]

      8. *-commutative N/A

         \[\leadsto x + \color{blue}{\left(\left(y - z\right) \cdot \frac{1}{a - z}\right)} \cdot \left(t - x\right) \]

      9. mult-flip-rev N/A

         \[\leadsto x + \color{blue}{\frac{y - z}{a - z}} \cdot \left(t - x\right) \]

      10. frac-2neg N/A

          \[\leadsto x + \color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      11. lift--.f64 N/A

          \[\leadsto x + \frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      12. sub-negate-rev N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      13. lower-/.f64 N/A

          \[\leadsto x + \color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}} \cdot \left(t - x\right) \]

      14. lower--.f64 N/A

          \[\leadsto x + \frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(t - x\right) \]

      15. lift--.f64 N/A

          \[\leadsto x + \frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)} \cdot \left(t - x\right) \]

      16. sub-negate-rev N/A

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

      17. lower--.f64 83.9

          \[\leadsto x + \frac{z - y}{\color{blue}{z - a}} \cdot \left(t - x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto x + \color{blue}{\frac{z - y}{z - a} \cdot \left(t - x\right)} \]

   4. Taylor expanded in z around 0

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a}} \]
   5. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{\color{blue}{a}} \]

      2. lower-*.f64 N/A

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a} \]

      3. lower--.f64 44.1

         \[\leadsto x + \frac{y \cdot \left(t - x\right)}{a} \]

   6. Applied rewrites 44.1%

      \[\leadsto x + \color{blue}{\frac{y \cdot \left(t - x\right)}{a}} \]

   7. Taylor expanded in x around 0

      \[\leadsto x + \frac{y \cdot t}{a} \]
   8. Step-by-step derivation

   9. Applied rewrites 37.8%

      \[\leadsto x + \frac{y \cdot t}{a} \]

   if -1.06e-40 < a < 1.19999999999999994e-12

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   7. Taylor expanded in y around -inf

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      3. lower--.f64 23.5

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 45.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y \cdot \left(x - t\right)}{z}\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\frac{x \cdot \left(y - a\right)}{z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* y (- x t)) z)) (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-229)
       (+ t x)
       (if (<= t_2 0.0)
         (/ (* x (- y a)) z)
         (if (<= t_2 5e+306) (+ t x) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (x - t)) / z;
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-229) {
		tmp = t + x;
	} else if (t_2 <= 0.0) {
		tmp = (x * (y - a)) / z;
	} else if (t_2 <= 5e+306) {
		tmp = t + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y * (x - t)) / z;
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -1e-229) {
		tmp = t + x;
	} else if (t_2 <= 0.0) {
		tmp = (x * (y - a)) / z;
	} else if (t_2 <= 5e+306) {
		tmp = t + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y * (x - t)) / z
	t_2 = x + ((y - z) * ((t - x) / (a - z)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -1e-229:
		tmp = t + x
	elif t_2 <= 0.0:
		tmp = (x * (y - a)) / z
	elif t_2 <= 5e+306:
		tmp = t + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y * Float64(x - t)) / z)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-229)
		tmp = Float64(t + x);
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(x * Float64(y - a)) / z);
	elseif (t_2 <= 5e+306)
		tmp = Float64(t + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y * (x - t)) / z;
	t_2 = x + ((y - z) * ((t - x) / (a - z)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -1e-229)
		tmp = t + x;
	elseif (t_2 <= 0.0)
		tmp = (x * (y - a)) / z;
	elseif (t_2 <= 5e+306)
		tmp = t + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-229], N[(t + x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(x * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 5e+306], N[(t + x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -inf.0 or 4.99999999999999993e306 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. +-commutative N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + \color{blue}{t} \]

      3. lift-*.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      4. lift-/.f64 N/A

         \[\leadsto -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} + t \]

      5. mult-flip N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      6. lift--.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      7. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      8. lift-*.f64 N/A

         \[\leadsto -1 \cdot \left(\left(y \cdot \left(t - x\right) - a \cdot \left(t - x\right)\right) \cdot \frac{1}{z}\right) + t \]

      9. distribute-rgt-out-- N/A

         \[\leadsto -1 \cdot \left(\left(\left(t - x\right) \cdot \left(y - a\right)\right) \cdot \frac{1}{z}\right) + t \]

      10. associate-*l* N/A

          \[\leadsto -1 \cdot \left(\left(t - x\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right)\right) + t \]

      11. associate-*r* N/A

          \[\leadsto \left(-1 \cdot \left(t - x\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      12. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      13. lift--.f64 N/A

          \[\leadsto \left(\mathsf{neg}\left(\left(t - x\right)\right)\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      14. sub-negate-rev N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      15. lift--.f64 N/A

          \[\leadsto \left(x - t\right) \cdot \left(\left(y - a\right) \cdot \frac{1}{z}\right) + t \]

      16. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\left(y - a\right) \cdot \frac{1}{z}}, t\right) \]

   6. Applied rewrites 54.2%

      \[\leadsto \mathsf{fma}\left(x - t, \color{blue}{\frac{y - a}{z}}, t\right) \]

   7. Taylor expanded in y around -inf

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]
   8. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

      3. lower--.f64 23.5

         \[\leadsto \frac{y \cdot \left(x - t\right)}{z} \]

   9. Applied rewrites 23.5%

      \[\leadsto \frac{y \cdot \left(x - t\right)}{\color{blue}{z}} \]

   if -inf.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999993e306

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 20.2

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 20.2%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 2.8

         \[\leadsto x + -1 \cdot x \]

   7. Applied rewrites 2.8%

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + -1 \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot x + x} \]

      3. lower-+.f64 2.8

         \[\leadsto \color{blue}{-1 \cdot x + x} \]

      4. lift--.f64 N/A

         \[\leadsto -1 \cdot x + x \]

      5. lift--.f64 N/A

         \[\leadsto -1 \cdot x + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{-1} \cdot x + x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{-1} \cdot x + x \]

      8. frac-2neg-rev N/A

         \[\leadsto -1 \cdot x + x \]

      9. sub-negate-rev N/A

         \[\leadsto -1 \cdot x + x \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot x + x \]

      11. sub-negate-rev N/A

          \[\leadsto -1 \cdot x + x \]

      12. lift--.f64 2.8

          \[\leadsto -1 \cdot x + x \]

      13. lift-*.f64 N/A

          \[\leadsto -1 \cdot x + x \]

      14. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) + x \]

      15. lower-neg.f64 2.8

          \[\leadsto \left(-x\right) + x \]

   9. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\left(-x\right) + x} \]

   10. Taylor expanded in x around 0

       \[\leadsto t + x \]
   11. Step-by-step derivation

   12. Applied rewrites 35.4%

       \[\leadsto t + x \]

   if -1.00000000000000007e-229 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \frac{x \cdot \left(y - a\right)}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

      3. lower--.f64 20.2

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{x \cdot \left(y - a\right)}{\color{blue}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 15: 45.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot \left(y - a\right)}{z}\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (* x (- y a)) z)) (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-229)
       (+ t x)
       (if (<= t_2 0.0) t_1 (if (<= t_2 5e+306) (+ t x) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * (y - a)) / z;
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-229) {
		tmp = t + x;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+306) {
		tmp = t + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (x * (y - a)) / z;
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= -1e-229) {
		tmp = t + x;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 5e+306) {
		tmp = t + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (x * (y - a)) / z
	t_2 = x + ((y - z) * ((t - x) / (a - z)))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= -1e-229:
		tmp = t + x
	elif t_2 <= 0.0:
		tmp = t_1
	elif t_2 <= 5e+306:
		tmp = t + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x * Float64(y - a)) / z)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-229)
		tmp = Float64(t + x);
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e+306)
		tmp = Float64(t + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (x * (y - a)) / z;
	t_2 = x + ((y - z) * ((t - x) / (a - z)));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= -1e-229)
		tmp = t + x;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 5e+306)
		tmp = t + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-229], N[(t + x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 5e+306], N[(t + x), $MachinePrecision], t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -inf.0 or -1.00000000000000007e-229 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0 or 4.99999999999999993e306 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto t + \color{blue}{-1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      2. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \color{blue}{\frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

      3. lower-/.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{\color{blue}{z}} \]

      4. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      5. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      6. lower--.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      7. lower-*.f64 N/A

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

      8. lower--.f64 46.9

         \[\leadsto t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z} \]

   4. Applied rewrites 46.9%

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]

   5. Taylor expanded in x around -inf

      \[\leadsto \frac{x \cdot \left(y - a\right)}{\color{blue}{z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

      3. lower--.f64 20.2

         \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]

   7. Applied rewrites 20.2%

      \[\leadsto \frac{x \cdot \left(y - a\right)}{\color{blue}{z}} \]

   if -inf.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000007e-229 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999993e306

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 20.2

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 20.2%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 2.8

         \[\leadsto x + -1 \cdot x \]

   7. Applied rewrites 2.8%

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + -1 \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot x + x} \]

      3. lower-+.f64 2.8

         \[\leadsto \color{blue}{-1 \cdot x + x} \]

      4. lift--.f64 N/A

         \[\leadsto -1 \cdot x + x \]

      5. lift--.f64 N/A

         \[\leadsto -1 \cdot x + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{-1} \cdot x + x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{-1} \cdot x + x \]

      8. frac-2neg-rev N/A

         \[\leadsto -1 \cdot x + x \]

      9. sub-negate-rev N/A

         \[\leadsto -1 \cdot x + x \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot x + x \]

      11. sub-negate-rev N/A

          \[\leadsto -1 \cdot x + x \]

      12. lift--.f64 2.8

          \[\leadsto -1 \cdot x + x \]

      13. lift-*.f64 N/A

          \[\leadsto -1 \cdot x + x \]

      14. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) + x \]

      15. lower-neg.f64 2.8

          \[\leadsto \left(-x\right) + x \]

   9. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\left(-x\right) + x} \]

   10. Taylor expanded in x around 0

       \[\leadsto t + x \]
   11. Step-by-step derivation

   12. Applied rewrites 35.4%

       \[\leadsto t + x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 39.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ t_2 := x \cdot \frac{t}{x}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-229}:\\ \;\;\;\;t + x\\ \mathbf{elif}\;t\_1 \leq 10^{-182}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;t + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))) (t_2 (* x (/ t x))))
   (if (<= t_1 -1e-229)
     (+ t x)
     (if (<= t_1 1e-182) t_2 (if (<= t_1 5e+306) (+ t x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double t_2 = x * (t / x);
	double tmp;
	if (t_1 <= -1e-229) {
		tmp = t + x;
	} else if (t_1 <= 1e-182) {
		tmp = t_2;
	} else if (t_1 <= 5e+306) {
		tmp = t + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x + ((y - z) * ((t - x) / (a - z)))
    t_2 = x * (t / x)
    if (t_1 <= (-1d-229)) then
        tmp = t + x
    else if (t_1 <= 1d-182) then
        tmp = t_2
    else if (t_1 <= 5d+306) then
        tmp = t + x
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double t_2 = x * (t / x);
	double tmp;
	if (t_1 <= -1e-229) {
		tmp = t + x;
	} else if (t_1 <= 1e-182) {
		tmp = t_2;
	} else if (t_1 <= 5e+306) {
		tmp = t + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + ((y - z) * ((t - x) / (a - z)))
	t_2 = x * (t / x)
	tmp = 0
	if t_1 <= -1e-229:
		tmp = t + x
	elif t_1 <= 1e-182:
		tmp = t_2
	elif t_1 <= 5e+306:
		tmp = t + x
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	t_2 = Float64(x * Float64(t / x))
	tmp = 0.0
	if (t_1 <= -1e-229)
		tmp = Float64(t + x);
	elseif (t_1 <= 1e-182)
		tmp = t_2;
	elseif (t_1 <= 5e+306)
		tmp = Float64(t + x);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + ((y - z) * ((t - x) / (a - z)));
	t_2 = x * (t / x);
	tmp = 0.0;
	if (t_1 <= -1e-229)
		tmp = t + x;
	elseif (t_1 <= 1e-182)
		tmp = t_2;
	elseif (t_1 <= 5e+306)
		tmp = t + x;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(t / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-229], N[(t + x), $MachinePrecision], If[LessEqual[t$95$1, 1e-182], t$95$2, If[LessEqual[t$95$1, 5e+306], N[(t + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -1.00000000000000007e-229 or 1e-182 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999993e306

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]
   3. Step-by-step derivation

      1. lower--.f64 20.2

         \[\leadsto x + \left(t - \color{blue}{x}\right) \]

   4. Applied rewrites 20.2%

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   6. Step-by-step derivation

      1. lower-*.f64 2.8

         \[\leadsto x + -1 \cdot x \]

   7. Applied rewrites 2.8%

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   8. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + -1 \cdot x} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot x + x} \]

      3. lower-+.f64 2.8

         \[\leadsto \color{blue}{-1 \cdot x + x} \]

      4. lift--.f64 N/A

         \[\leadsto -1 \cdot x + x \]

      5. lift--.f64 N/A

         \[\leadsto -1 \cdot x + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{-1} \cdot x + x \]

      7. associate-*l/ N/A

         \[\leadsto \color{blue}{-1} \cdot x + x \]

      8. frac-2neg-rev N/A

         \[\leadsto -1 \cdot x + x \]

      9. sub-negate-rev N/A

         \[\leadsto -1 \cdot x + x \]

      10. lift--.f64 N/A

          \[\leadsto -1 \cdot x + x \]

      11. sub-negate-rev N/A

          \[\leadsto -1 \cdot x + x \]

      12. lift--.f64 2.8

          \[\leadsto -1 \cdot x + x \]

      13. lift-*.f64 N/A

          \[\leadsto -1 \cdot x + x \]

      14. mul-1-neg N/A

          \[\leadsto \left(\mathsf{neg}\left(x\right)\right) + x \]

      15. lower-neg.f64 2.8

          \[\leadsto \left(-x\right) + x \]

   9. Applied rewrites 2.8%

      \[\leadsto \color{blue}{\left(-x\right) + x} \]

   10. Taylor expanded in x around 0

       \[\leadsto t + x \]
   11. Step-by-step derivation

   12. Applied rewrites 35.4%

       \[\leadsto t + x \]

   if -1.00000000000000007e-229 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 1e-182 or 4.99999999999999993e306 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 79.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{x + \left(y - z\right) \cdot \frac{t - x}{a - z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + x} \]

      3. add-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} - \left(\mathsf{neg}\left(x\right)\right)} \]

      4. sub-flip N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right)} \]

      5. lift-*.f64 N/A

         \[\leadsto \color{blue}{\left(y - z\right) \cdot \frac{t - x}{a - z}} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      6. *-commutative N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z} \cdot \left(y - z\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{t - x}{a - z}} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      8. mult-flip N/A

         \[\leadsto \color{blue}{\left(\left(t - x\right) \cdot \frac{1}{a - z}\right)} \cdot \left(y - z\right) + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      9. associate-*l* N/A

         \[\leadsto \color{blue}{\left(t - x\right) \cdot \left(\frac{1}{a - z} \cdot \left(y - z\right)\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right)} + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) \]

      11. remove-double-neg N/A

          \[\leadsto \left(\frac{1}{a - z} \cdot \left(y - z\right)\right) \cdot \left(t - x\right) + \color{blue}{x} \]

      12. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{a - z} \cdot \left(y - z\right), t - x, x\right)} \]

      13. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(y - z\right) \cdot \frac{1}{a - z}}, t - x, x\right) \]

      14. mult-flip-rev N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y - z}{a - z}}, t - x, x\right) \]

      15. frac-2neg N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{\mathsf{neg}\left(\left(y - z\right)\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      16. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\mathsf{neg}\left(\color{blue}{\left(y - z\right)}\right)}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      17. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      18. lower-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{z - y}{\mathsf{neg}\left(\left(a - z\right)\right)}}, t - x, x\right) \]

      19. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{z - y}}{\mathsf{neg}\left(\left(a - z\right)\right)}, t - x, x\right) \]

      20. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\mathsf{neg}\left(\color{blue}{\left(a - z\right)}\right)}, t - x, x\right) \]

      21. sub-negate-rev N/A

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

      22. lower--.f64 83.9

          \[\leadsto \mathsf{fma}\left(\frac{z - y}{\color{blue}{z - a}}, t - x, x\right) \]

   3. Applied rewrites 83.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)} \]

   4. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x \cdot \left(1 + \left(-1 \cdot \frac{z - y}{z - a} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right)} \]
   5. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x \cdot \color{blue}{\left(1 + \left(-1 \cdot \frac{z - y}{z - a} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot \left(1 + \color{blue}{\left(-1 \cdot \frac{z - y}{z - a} + \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)}\right) \]

      3. lower-fma.f64 N/A

         \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \color{blue}{\frac{z - y}{z - a}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      4. lower-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{z - a}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      5. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{\color{blue}{z} - a}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      6. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{z - \color{blue}{a}}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      7. lower-/.f64 N/A

         \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{z - a}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      8. lower-*.f64 N/A

         \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{z - a}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      9. lower--.f64 N/A

         \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{z - a}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      10. lower-*.f64 N/A

          \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{z - a}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

      11. lower--.f64 64.3

          \[\leadsto x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{z - a}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right) \]

   6. Applied rewrites 64.3%

      \[\leadsto \color{blue}{x \cdot \left(1 + \mathsf{fma}\left(-1, \frac{z - y}{z - a}, \frac{t \cdot \left(z - y\right)}{x \cdot \left(z - a\right)}\right)\right)} \]

   7. Taylor expanded in z around inf

      \[\leadsto x \cdot \frac{t}{\color{blue}{x}} \]
   8. Step-by-step derivation

      1. lower-/.f64 22.8

         \[\leadsto x \cdot \frac{t}{x} \]

   9. Applied rewrites 22.8%

      \[\leadsto x \cdot \frac{t}{\color{blue}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 35.4% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ t + x \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ t x))

C

double code(double x, double y, double z, double t, double a) {
	return t + x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = t + x
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return t + x;
}

Python

def code(x, y, z, t, a):
	return t + x

Julia

function code(x, y, z, t, a)
	return Float64(t + x)
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = t + x;
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(t + x), $MachinePrecision]

Derivation

1. Initial program 79.5%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]

2. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
3. Step-by-step derivation

   1. lower--.f64 20.2

      \[\leadsto x + \left(t - \color{blue}{x}\right) \]

4. Applied rewrites 20.2%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Taylor expanded in x around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
6. Step-by-step derivation

   1. lower-*.f64 2.8

      \[\leadsto x + -1 \cdot x \]

7. Applied rewrites 2.8%

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
8. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{x + -1 \cdot x} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{-1 \cdot x + x} \]

   3. lower-+.f64 2.8

      \[\leadsto \color{blue}{-1 \cdot x + x} \]

   4. lift--.f64 N/A

      \[\leadsto -1 \cdot x + x \]

   5. lift--.f64 N/A

      \[\leadsto -1 \cdot x + x \]

   6. associate-*r/ N/A

      \[\leadsto \color{blue}{-1} \cdot x + x \]

   7. associate-*l/ N/A

      \[\leadsto \color{blue}{-1} \cdot x + x \]

   8. frac-2neg-rev N/A

      \[\leadsto -1 \cdot x + x \]

   9. sub-negate-rev N/A

      \[\leadsto -1 \cdot x + x \]

   10. lift--.f64 N/A

       \[\leadsto -1 \cdot x + x \]

   11. sub-negate-rev N/A

       \[\leadsto -1 \cdot x + x \]

   12. lift--.f64 2.8

       \[\leadsto -1 \cdot x + x \]

   13. lift-*.f64 N/A

       \[\leadsto -1 \cdot x + x \]

   14. mul-1-neg N/A

       \[\leadsto \left(\mathsf{neg}\left(x\right)\right) + x \]

   15. lower-neg.f64 2.8

       \[\leadsto \left(-x\right) + x \]

9. Applied rewrites 2.8%

   \[\leadsto \color{blue}{\left(-x\right) + x} \]

10. Taylor expanded in x around 0

    \[\leadsto t + x \]
11. Step-by-step derivation

12. Applied rewrites 35.4%

    \[\leadsto t + x \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2025156 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))