Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.1% → 91.7%

Time: 6.7s

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: 80.1% 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: 91.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \left(y - z\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z} - \frac{x}{a - z}, y - z, x\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t + \left(-x\right) \cdot \frac{a - y}{z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \left(\left(1 - \frac{y - z}{a - z}\right) + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (* (- y z) t_1))))
   (if (<= t_2 -5e-234)
     (fma (- (/ t (- a z)) (/ x (- a z))) (- y z) x)
     (if (<= t_2 0.0)
       (+ t (* (- x) (/ (- a y) z)))
       (if (<= t_2 5e+17)
         (* x (+ (- 1.0 (/ (- y z) (- a z))) (/ (* t (- y z)) (* x (- a z)))))
         (fma t_1 (- y z) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / (a - z);
	double t_2 = x + ((y - z) * t_1);
	double tmp;
	if (t_2 <= -5e-234) {
		tmp = fma(((t / (a - z)) - (x / (a - z))), (y - z), x);
	} else if (t_2 <= 0.0) {
		tmp = t + (-x * ((a - y) / z));
	} else if (t_2 <= 5e+17) {
		tmp = x * ((1.0 - ((y - z) / (a - z))) + ((t * (y - z)) / (x * (a - z))));
	} else {
		tmp = fma(t_1, (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / Float64(a - z))
	t_2 = Float64(x + Float64(Float64(y - z) * t_1))
	tmp = 0.0
	if (t_2 <= -5e-234)
		tmp = fma(Float64(Float64(t / Float64(a - z)) - Float64(x / Float64(a - z))), Float64(y - z), x);
	elseif (t_2 <= 0.0)
		tmp = Float64(t + Float64(Float64(-x) * Float64(Float64(a - y) / z)));
	elseif (t_2 <= 5e+17)
		tmp = Float64(x * Float64(Float64(1.0 - Float64(Float64(y - z) / Float64(a - z))) + Float64(Float64(t * Float64(y - z)) / Float64(x * Float64(a - z)))));
	else
		tmp = fma(t_1, Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-234], N[(N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t + N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+17], N[(x * N[(N[(1.0 - N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(x * N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -4.99999999999999979e-234

   1. Initial program 92.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 92.9

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

   4. Applied rewrites 92.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 92.9

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

   6. Applied rewrites 92.9%

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

   if -4.99999999999999979e-234 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 6.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 7.6

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 83.5

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.8

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

   10. Applied rewrites 98.8%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 5e17

   1. Initial program 77.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. associate-+r+ N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. sub-div N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. sub-div N/A

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

      9. lower-/.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 97.9%

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

   if 5e17 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 94.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 94.7

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

   4. Applied rewrites 94.7%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z} - \frac{x}{a - z}, y - z, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;t + \left(-x\right) \cdot \frac{a - y}{z}\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 5 \cdot 10^{+17}:\\ \;\;\;\;x \cdot \left(\left(1 - \frac{y - z}{a - z}\right) + \frac{t \cdot \left(y - z\right)}{x \cdot \left(a - z\right)}\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \left(y - z\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-234} \lor \neg \left(t\_2 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(-x\right) \cdot \frac{a - y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (* (- y z) t_1))))
   (if (or (<= t_2 -5e-234) (not (<= t_2 0.0)))
     (fma t_1 (- y z) x)
     (+ t (* (- x) (/ (- a y) z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / (a - z);
	double t_2 = x + ((y - z) * t_1);
	double tmp;
	if ((t_2 <= -5e-234) || !(t_2 <= 0.0)) {
		tmp = fma(t_1, (y - z), x);
	} else {
		tmp = t + (-x * ((a - y) / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / Float64(a - z))
	t_2 = Float64(x + Float64(Float64(y - z) * t_1))
	tmp = 0.0
	if ((t_2 <= -5e-234) || !(t_2 <= 0.0))
		tmp = fma(t_1, Float64(y - z), x);
	else
		tmp = Float64(t + Float64(Float64(-x) * Float64(Float64(a - y) / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e-234], N[Not[LessEqual[t$95$2, 0.0]], $MachinePrecision]], N[(t$95$1 * N[(y - z), $MachinePrecision] + x), $MachinePrecision], N[(t + N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -4.99999999999999979e-234 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 90.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 90.8

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

   4. Applied rewrites 90.8%

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

   if -4.99999999999999979e-234 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 6.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 7.6

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 83.5

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.8

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

   10. Applied rewrites 98.8%

       \[\leadsto t + -1 \cdot \color{blue}{\left(x \cdot \frac{a - y}{z}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5 \cdot 10^{-234} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \left(-x\right) \cdot \frac{a - y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \left(y - z\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-234} \lor \neg \left(t\_2 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t\_1, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot \left(y - a\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (* (- y z) t_1))))
   (if (or (<= t_2 -5e-234) (not (<= t_2 0.0)))
     (fma t_1 (- y z) x)
     (+ t (/ (* x (- y a)) z)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / (a - z);
	double t_2 = x + ((y - z) * t_1);
	double tmp;
	if ((t_2 <= -5e-234) || !(t_2 <= 0.0)) {
		tmp = fma(t_1, (y - z), x);
	} else {
		tmp = t + ((x * (y - a)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / Float64(a - z))
	t_2 = Float64(x + Float64(Float64(y - z) * t_1))
	tmp = 0.0
	if ((t_2 <= -5e-234) || !(t_2 <= 0.0))
		tmp = fma(t_1, Float64(y - z), x);
	else
		tmp = Float64(t + Float64(Float64(x * Float64(y - a)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$2, -5e-234], N[Not[LessEqual[t$95$2, 0.0]], $MachinePrecision]], N[(t$95$1 * N[(y - z), $MachinePrecision] + x), $MachinePrecision], N[(t + N[(N[(x * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -4.99999999999999979e-234 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 90.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 90.8

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

   4. Applied rewrites 90.8%

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

   if -4.99999999999999979e-234 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 6.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 7.6

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 83.5

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 83.5

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

   10. Applied rewrites 83.5%

       \[\leadsto t + \frac{x \cdot \left(y - a\right)}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5 \cdot 10^{-234} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t + \frac{x \cdot \left(y - a\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \left(y - z\right) \cdot t\_1\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z} - \frac{x}{a - z}, y - z, x\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t + \left(-x\right) \cdot \frac{a - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (* (- y z) t_1))))
   (if (<= t_2 -5e-234)
     (fma (- (/ t (- a z)) (/ x (- a z))) (- y z) x)
     (if (<= t_2 0.0) (+ t (* (- x) (/ (- a y) z))) (fma t_1 (- y z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / (a - z);
	double t_2 = x + ((y - z) * t_1);
	double tmp;
	if (t_2 <= -5e-234) {
		tmp = fma(((t / (a - z)) - (x / (a - z))), (y - z), x);
	} else if (t_2 <= 0.0) {
		tmp = t + (-x * ((a - y) / z));
	} else {
		tmp = fma(t_1, (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / Float64(a - z))
	t_2 = Float64(x + Float64(Float64(y - z) * t_1))
	tmp = 0.0
	if (t_2 <= -5e-234)
		tmp = fma(Float64(Float64(t / Float64(a - z)) - Float64(x / Float64(a - z))), Float64(y - z), x);
	elseif (t_2 <= 0.0)
		tmp = Float64(t + Float64(Float64(-x) * Float64(Float64(a - y) / z)));
	else
		tmp = fma(t_1, Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-234], N[(N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(x / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(t + N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$1 * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -4.99999999999999979e-234

   1. Initial program 92.7%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 92.9

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

   4. Applied rewrites 92.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-div N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 92.9

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

   6. Applied rewrites 92.9%

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

   if -4.99999999999999979e-234 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 6.4%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 7.6

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

   4. Applied rewrites 7.6%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 83.5

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

   7. Applied rewrites 83.5%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.8

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

   10. Applied rewrites 98.8%

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

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

   1. Initial program 88.8%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 88.9

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

   4. Applied rewrites 88.9%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -5 \cdot 10^{-234}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a - z} - \frac{x}{a - z}, y - z, x\right)\\ \mathbf{elif}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0:\\ \;\;\;\;t + \left(-x\right) \cdot \frac{a - y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-78}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -9e-90)
   (fma (- t x) (/ (- y z) a) x)
   (if (<= a 6e-78)
     (- t (/ (* y (- t x)) z))
     (if (<= a 1.35e+82) (* t (/ (- y z) (- a z))) (fma y (/ (- t x) a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9e-90) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else if (a <= 6e-78) {
		tmp = t - ((y * (t - x)) / z);
	} else if (a <= 1.35e+82) {
		tmp = t * ((y - z) / (a - z));
	} else {
		tmp = fma(y, ((t - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9e-90)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	elseif (a <= 6e-78)
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	elseif (a <= 1.35e+82)
		tmp = Float64(t * Float64(Float64(y - z) / Float64(a - z)));
	else
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9e-90], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 6e-78], N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.35e+82], N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if a < -9.00000000000000017e-90

   1. Initial program 80.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -9.00000000000000017e-90 < a < 5.99999999999999975e-78

   1. Initial program 69.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 86.5

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

   8. Applied rewrites 86.5%

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

   if 5.99999999999999975e-78 < a < 1.35e82

   1. Initial program 78.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 79.1

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

   4. Applied rewrites 79.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 72.3

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

   7. Applied rewrites 72.3%

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

   if 1.35e82 < a

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 84.3

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

   5. Applied rewrites 84.3%

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

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-78}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{elif}\;a \leq 1.35 \cdot 10^{+82}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 66.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -2.35 \cdot 10^{+148}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-68}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t - x}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- t x) a) x)))
   (if (<= a -2.35e+148)
     t_1
     (if (<= a -4e-68)
       (* t (/ (- y z) (- a z)))
       (if (<= a 5.8e+80) (fma (- y) (/ (- t x) z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((t - x) / a), x);
	double tmp;
	if (a <= -2.35e+148) {
		tmp = t_1;
	} else if (a <= -4e-68) {
		tmp = t * ((y - z) / (a - z));
	} else if (a <= 5.8e+80) {
		tmp = fma(-y, ((t - x) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -2.35e+148)
		tmp = t_1;
	elseif (a <= -4e-68)
		tmp = Float64(t * Float64(Float64(y - z) / Float64(a - z)));
	elseif (a <= 5.8e+80)
		tmp = fma(Float64(-y), Float64(Float64(t - x) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.35e+148], t$95$1, If[LessEqual[a, -4e-68], N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 5.8e+80], N[((-y) * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.3499999999999999e148 or 5.79999999999999971e80 < a

   1. Initial program 94.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 80.1

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

   5. Applied rewrites 80.1%

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

   if -2.3499999999999999e148 < a < -4.00000000000000027e-68

   1. Initial program 64.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 64.9

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

   4. Applied rewrites 64.9%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 56.9

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

   7. Applied rewrites 56.9%

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

   if -4.00000000000000027e-68 < a < 5.79999999999999971e80

   1. Initial program 73.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 63.5

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

   5. Applied rewrites 63.5%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 46.2

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

   8. Applied rewrites 46.2%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 N/A

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

       6. sub-div N/A

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

       7. lower-/.f64 N/A

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

       8. lift--.f64 77.6

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

   11. Applied rewrites 77.6%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.35 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq -4 \cdot 10^{-68}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t - x}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -4.8 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -0.088:\\ \;\;\;\;\left(-x\right) \cdot \frac{a - y}{z}\\ \mathbf{elif}\;a \leq 400000:\\ \;\;\;\;\left(-t\right) \cdot \left(\frac{y}{z} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- t x) a) x)))
   (if (<= a -4.8e+119)
     t_1
     (if (<= a -0.088)
       (* (- x) (/ (- a y) z))
       (if (<= a 400000.0) (* (- t) (- (/ y z) 1.0)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, ((t - x) / a), x);
	double tmp;
	if (a <= -4.8e+119) {
		tmp = t_1;
	} else if (a <= -0.088) {
		tmp = -x * ((a - y) / z);
	} else if (a <= 400000.0) {
		tmp = -t * ((y / z) - 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -4.8e+119)
		tmp = t_1;
	elseif (a <= -0.088)
		tmp = Float64(Float64(-x) * Float64(Float64(a - y) / z));
	elseif (a <= 400000.0)
		tmp = Float64(Float64(-t) * Float64(Float64(y / z) - 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -4.8e+119], t$95$1, If[LessEqual[a, -0.088], N[((-x) * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 400000.0], N[((-t) * N[(N[(y / z), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.8e119 or 4e5 < a

   1. Initial program 93.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 73.3

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

   5. Applied rewrites 73.3%

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

   if -4.8e119 < a < -0.087999999999999995

   1. Initial program 51.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 52.2

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

   4. Applied rewrites 52.2%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 41.8

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

   7. Applied rewrites 41.8%

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

   8. Taylor expanded in x around -inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 50.7

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

   10. Applied rewrites 50.7%

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

       1. lift-*.f64 N/A

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

       2. lift-*.f64 N/A

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

       3. lift--.f64 N/A

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

       4. lift-/.f64 N/A

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

       5. associate-*r* N/A

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

       6. mul-1-neg N/A

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

       7. lower-*.f64 N/A

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

       8. lower-neg.f64 N/A

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

       9. lift-/.f64 N/A

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

       10. lift--.f64 50.7

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

   12. Applied rewrites 50.7%

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

   if -0.087999999999999995 < a < 4e5

   1. Initial program 72.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 62.2

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

   5. Applied rewrites 62.2%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 55.9

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

   8. Applied rewrites 55.9%

      \[\leadsto -t \cdot \left(\frac{y}{z} - 1\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+119}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq -0.088:\\ \;\;\;\;\left(-x\right) \cdot \frac{a - y}{z}\\ \mathbf{elif}\;a \leq 400000:\\ \;\;\;\;\left(-t\right) \cdot \left(\frac{y}{z} - 1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-91}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.1e-91)
   (+ x (* (- y z) (/ t (- a z))))
   (if (<= a 9.5e-44)
     (- t (/ (* y (- t x)) z))
     (+ x (* y (/ (- t x) (- a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.1e-91) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else if (a <= 9.5e-44) {
		tmp = t - ((y * (t - x)) / z);
	} else {
		tmp = x + (y * ((t - x) / (a - z)));
	}
	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) :: tmp
    if (a <= (-4.1d-91)) then
        tmp = x + ((y - z) * (t / (a - z)))
    else if (a <= 9.5d-44) then
        tmp = t - ((y * (t - x)) / z)
    else
        tmp = x + (y * ((t - x) / (a - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.1e-91) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else if (a <= 9.5e-44) {
		tmp = t - ((y * (t - x)) / z);
	} else {
		tmp = x + (y * ((t - x) / (a - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -4.1e-91:
		tmp = x + ((y - z) * (t / (a - z)))
	elif a <= 9.5e-44:
		tmp = t - ((y * (t - x)) / z)
	else:
		tmp = x + (y * ((t - x) / (a - z)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.1e-91)
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	elseif (a <= 9.5e-44)
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(t - x) / Float64(a - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -4.1e-91)
		tmp = x + ((y - z) * (t / (a - z)));
	elseif (a <= 9.5e-44)
		tmp = t - ((y * (t - x)) / z);
	else
		tmp = x + (y * ((t - x) / (a - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.1e-91], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.5e-44], N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -4.10000000000000024e-91

   1. Initial program 80.5%

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

   3. Taylor expanded in x around 0

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

   5. Applied rewrites 72.2%

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

   if -4.10000000000000024e-91 < a < 9.49999999999999924e-44

   1. Initial program 69.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 85.9

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

   8. Applied rewrites 85.9%

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

   if 9.49999999999999924e-44 < a

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{y} \cdot \frac{t - x}{a - z} \]
   4. Step-by-step derivation

   5. Applied rewrites 81.5%

      \[\leadsto x + \color{blue}{y} \cdot \frac{t - x}{a - z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.1 \cdot 10^{-91}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -9e-90)
   (fma (- t x) (/ (- y z) a) x)
   (if (<= a 9.5e-44)
     (- t (/ (* y (- t x)) z))
     (+ x (* y (/ (- t x) (- a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9e-90) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else if (a <= 9.5e-44) {
		tmp = t - ((y * (t - x)) / z);
	} else {
		tmp = x + (y * ((t - x) / (a - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9e-90)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	elseif (a <= 9.5e-44)
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(t - x) / Float64(a - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9e-90], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 9.5e-44], N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.00000000000000017e-90

   1. Initial program 80.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -9.00000000000000017e-90 < a < 9.49999999999999924e-44

   1. Initial program 69.7%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 68.0

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

   5. Applied rewrites 68.0%

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

   6. Taylor expanded in z around inf

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

      1. lower-+.f64 N/A

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

      2. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 85.9

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

   8. Applied rewrites 85.9%

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

   if 9.49999999999999924e-44 < a

   1. Initial program 91.0%

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

   3. Taylor expanded in y around inf

      \[\leadsto x + \color{blue}{y} \cdot \frac{t - x}{a - z} \]
   4. Step-by-step derivation

   5. Applied rewrites 81.5%

      \[\leadsto x + \color{blue}{y} \cdot \frac{t - x}{a - z} \]
3. Recombined 3 regimes into one program.

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-44}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \frac{t - x}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+119} \lor \neg \left(a \leq 5.8 \cdot 10^{+80}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t - x}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.8e+119) (not (<= a 5.8e+80)))
   (fma y (/ (- t x) a) x)
   (fma (- y) (/ (- t x) z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.8e+119) || !(a <= 5.8e+80)) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = fma(-y, ((t - x) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.8e+119) || !(a <= 5.8e+80))
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = fma(Float64(-y), Float64(Float64(t - x) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.8e+119], N[Not[LessEqual[a, 5.8e+80]], $MachinePrecision]], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[((-y) * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.8e119 or 5.79999999999999971e80 < a

   1. Initial program 95.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 78.8

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

   5. Applied rewrites 78.8%

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

   if -4.8e119 < a < 5.79999999999999971e80

   1. Initial program 70.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 39.7

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

   8. Applied rewrites 39.7%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 N/A

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

       6. sub-div N/A

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

       7. lower-/.f64 N/A

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

       8. lift--.f64 70.9

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

   11. Applied rewrites 70.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.8 \cdot 10^{+119} \lor \neg \left(a \leq 5.8 \cdot 10^{+80}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t - x}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 51.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+82}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{+58}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 3.9 \cdot 10^{+157}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e+82)
   t
   (if (<= z 1.16e+58)
     (fma y (/ t a) x)
     (if (<= z 3.9e+157) (/ (* x y) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e+82) {
		tmp = t;
	} else if (z <= 1.16e+58) {
		tmp = fma(y, (t / a), x);
	} else if (z <= 3.9e+157) {
		tmp = (x * y) / z;
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e+82)
		tmp = t;
	elseif (z <= 1.16e+58)
		tmp = fma(y, Float64(t / a), x);
	elseif (z <= 3.9e+157)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e+82], t, If[LessEqual[z, 1.16e+58], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.9e+157], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e82 or 3.89999999999999971e157 < z

   1. Initial program 58.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 41.7%

      \[\leadsto \color{blue}{t} \]

   if -1.35e82 < z < 1.1600000000000001e58

   1. Initial program 89.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 67.1

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

   5. Applied rewrites 67.1%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.8%

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

   if 1.1600000000000001e58 < z < 3.89999999999999971e157

   1. Initial program 84.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{x \cdot y}{\color{blue}{z}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{x \cdot y}{z} \]

      2. lower-*.f64 40.8

         \[\leadsto \frac{x \cdot y}{z} \]

   8. Applied rewrites 40.8%

      \[\leadsto \frac{x \cdot y}{\color{blue}{z}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 12: 69.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -9 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;a \leq 5.8 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{t - x}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -9e-90)
   (fma (- t x) (/ (- y z) a) x)
   (if (<= a 5.8e+80) (fma (- y) (/ (- t x) z) t) (fma y (/ (- t x) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -9e-90) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else if (a <= 5.8e+80) {
		tmp = fma(-y, ((t - x) / z), t);
	} else {
		tmp = fma(y, ((t - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -9e-90)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	elseif (a <= 5.8e+80)
		tmp = fma(Float64(-y), Float64(Float64(t - x) / z), t);
	else
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -9e-90], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 5.8e+80], N[((-y) * N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.00000000000000017e-90

   1. Initial program 80.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 68.1

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

   5. Applied rewrites 68.1%

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

   if -9.00000000000000017e-90 < a < 5.79999999999999971e80

   1. Initial program 72.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 64.3

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 47.1

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

   8. Applied rewrites 47.1%

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

   9. Taylor expanded in y around 0

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

       1. +-commutative N/A

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

       2. associate-*r* N/A

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

       3. lower-fma.f64 N/A

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

       4. mul-1-neg N/A

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

       5. lower-neg.f64 N/A

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

       6. sub-div N/A

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

       7. lower-/.f64 N/A

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

       8. lift--.f64 79.0

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

   11. Applied rewrites 79.0%

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

   if 5.79999999999999971e80 < a

   1. Initial program 95.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 84.3

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

   5. Applied rewrites 84.3%

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

Alternative 13: 59.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.7 \cdot 10^{-12} \lor \neg \left(a \leq 400000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \left(\frac{y}{z} - 1\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.7e-12) (not (<= a 400000.0)))
   (fma y (/ (- t x) a) x)
   (* (- t) (- (/ y z) 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.7e-12) || !(a <= 400000.0)) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = -t * ((y / z) - 1.0);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.7e-12) || !(a <= 400000.0))
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = Float64(Float64(-t) * Float64(Float64(y / z) - 1.0));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.7e-12], N[Not[LessEqual[a, 400000.0]], $MachinePrecision]], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[((-t) * N[(N[(y / z), $MachinePrecision] - 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.7000000000000003e-12 or 4e5 < a

   1. Initial program 86.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 65.2

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

   5. Applied rewrites 65.2%

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

   if -5.7000000000000003e-12 < a < 4e5

   1. Initial program 71.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lift--.f64 63.0

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

   5. Applied rewrites 63.0%

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

   6. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 55.7

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

   8. Applied rewrites 55.7%

      \[\leadsto -t \cdot \left(\frac{y}{z} - 1\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 60.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.7 \cdot 10^{-12} \lor \neg \left(a \leq 400000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(-t\right) \cdot \left(\frac{y}{z} - 1\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 56.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-93} \lor \neg \left(a \leq 5.5 \cdot 10^{-89}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -5.2e-93) (not (<= a 5.5e-89)))
   (fma y (/ (- t x) a) x)
   (* y (/ (- x t) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -5.2e-93) || !(a <= 5.5e-89)) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = y * ((x - t) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -5.2e-93) || !(a <= 5.5e-89))
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = Float64(y * Float64(Float64(x - t) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -5.2e-93], N[Not[LessEqual[a, 5.5e-89]], $MachinePrecision]], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -5.1999999999999997e-93 or 5.50000000000000012e-89 < a

   1. Initial program 84.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 60.0

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

   5. Applied rewrites 60.0%

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

   if -5.1999999999999997e-93 < a < 5.50000000000000012e-89

   1. Initial program 70.2%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 70.4

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

   4. Applied rewrites 70.4%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 88.8

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

   7. Applied rewrites 88.8%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

         \[\leadsto y \cdot \frac{x - t}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \frac{x - t}{z} \]

      4. lower--.f64 52.9

         \[\leadsto y \cdot \frac{x - t}{z} \]

   10. Applied rewrites 52.9%

       \[\leadsto y \cdot \color{blue}{\frac{x - t}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.2 \cdot 10^{-93} \lor \neg \left(a \leq 5.5 \cdot 10^{-89}\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 50.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-91} \lor \neg \left(a \leq 2300000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -4.6e-91) (not (<= a 2300000.0)))
   (fma y (/ t a) x)
   (* y (/ (- x t) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -4.6e-91) || !(a <= 2300000.0)) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = y * ((x - t) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -4.6e-91) || !(a <= 2300000.0))
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(y * Float64(Float64(x - t) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -4.6e-91], N[Not[LessEqual[a, 2300000.0]], $MachinePrecision]], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -4.59999999999999991e-91 or 2.3e6 < a

   1. Initial program 85.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 64.0

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

   5. Applied rewrites 64.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 57.9%

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

   if -4.59999999999999991e-91 < a < 2.3e6

   1. Initial program 70.9%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. +-commutative N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lift-/.f64 N/A

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

      11. lift--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. lift--.f64 71.1

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

   4. Applied rewrites 71.1%

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

   5. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{t + -1 \cdot \frac{y \cdot \left(t - x\right) - a \cdot \left(t - x\right)}{z}} \]
   6. 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. mul-1-neg N/A

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

      3. lower-neg.f64 N/A

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

      4. lower-/.f64 N/A

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

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

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower--.f64 85.8

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

   7. Applied rewrites 85.8%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

         \[\leadsto y \cdot \frac{x - t}{z} \]

      3. lower-/.f64 N/A

         \[\leadsto y \cdot \frac{x - t}{z} \]

      4. lower--.f64 49.0

         \[\leadsto y \cdot \frac{x - t}{z} \]

   10. Applied rewrites 49.0%

       \[\leadsto y \cdot \color{blue}{\frac{x - t}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{-91} \lor \neg \left(a \leq 2300000\right):\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 37.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1150000:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{+70}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1150000.0) x (if (<= a 4.2e+70) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1150000.0) {
		tmp = x;
	} else if (a <= 4.2e+70) {
		tmp = t;
	} else {
		tmp = x;
	}
	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) :: tmp
    if (a <= (-1150000.0d0)) then
        tmp = x
    else if (a <= 4.2d+70) then
        tmp = t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1150000.0) {
		tmp = x;
	} else if (a <= 4.2e+70) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1150000.0:
		tmp = x
	elif a <= 4.2e+70:
		tmp = t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1150000.0)
		tmp = x;
	elseif (a <= 4.2e+70)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1150000.0)
		tmp = x;
	elseif (a <= 4.2e+70)
		tmp = t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1150000.0], x, If[LessEqual[a, 4.2e+70], t, x]]

Derivation

1. Split input into 2 regimes

2. if a < -1.15e6 or 4.20000000000000015e70 < a

   1. Initial program 88.5%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Applied rewrites 49.2%

      \[\leadsto \color{blue}{x} \]

   if -1.15e6 < a < 4.20000000000000015e70

   1. Initial program 72.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 35.7%

      \[\leadsto \color{blue}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 24.2% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 t)

C

double code(double x, double y, double z, double t, double a) {
	return t;
}

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
end function

Java

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

Python

def code(x, y, z, t, a):
	return t

Julia

function code(x, y, z, t, a)
	return t
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = t;
end

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 79.2%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{t} \]
4. Step-by-step derivation

5. Applied rewrites 23.6%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025051 
(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)))))