Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Percentage Accurate: 68.1% → 91.1%

Time: 5.4s

Alternatives: 18

Speedup: 0.9×

Specification

Math

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

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(Float64(y - z) * 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[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 68.1% accurate, 1.0× speedup

Math

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

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(Float64(y - z) * 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[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(y - z\right) \cdot t\_1 + x\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-290}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(-x \cdot \mathsf{fma}\left(-1, \frac{y - a}{z}, \frac{t \cdot \left(y - a\right)}{x \cdot z}\right)\right) + t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+278}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, t\_1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 (- INFINITY))
     (+ (* (- y z) t_1) x)
     (if (<= t_2 -1e-290)
       t_2
       (if (<= t_2 0.0)
         (+ (- (* x (fma -1.0 (/ (- y a) z) (/ (* t (- y a)) (* x z))))) t)
         (if (<= t_2 2e+278) t_2 (fma (- y z) t_1 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 - x)) / (a - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((y - z) * t_1) + x;
	} else if (t_2 <= -1e-290) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = -(x * fma(-1.0, ((y - a) / z), ((t * (y - a)) / (x * z)))) + t;
	} else if (t_2 <= 2e+278) {
		tmp = t_2;
	} else {
		tmp = fma((y - z), t_1, 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(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y - z) * t_1) + x);
	elseif (t_2 <= -1e-290)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-Float64(x * fma(-1.0, Float64(Float64(y - a) / z), Float64(Float64(t * Float64(y - a)) / Float64(x * z))))) + t);
	elseif (t_2 <= 2e+278)
		tmp = t_2;
	else
		tmp = fma(Float64(y - z), t_1, 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[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, -1e-290], t$95$2, If[LessEqual[t$95$2, 0.0], N[((-N[(x * N[(-1.0 * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + N[(N[(t * N[(y - a), $MachinePrecision]), $MachinePrecision] / N[(x * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) + t), $MachinePrecision], If[LessEqual[t$95$2, 2e+278], t$95$2, N[(N[(y - z), $MachinePrecision] * t$95$1 + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.0000000000000001e-290 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.99999999999999993e278

   1. Initial program 68.1%

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

   if -1.0000000000000001e-290 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. sub-div N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower-*.f64 N/A

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

      9. lift--.f64 N/A

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

      10. lower-*.f64 44.4

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

   7. Applied rewrites 44.4%

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

   if 1.99999999999999993e278 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

Alternative 2: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z}\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;\left(y - z\right) \cdot t\_1 + x\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-290}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(-\frac{\left(t - x\right) \cdot \left(y - a\right)}{z}\right) + t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+278}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, t\_1, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) (- a z))) (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 (- INFINITY))
     (+ (* (- y z) t_1) x)
     (if (<= t_2 -1e-290)
       t_2
       (if (<= t_2 0.0)
         (+ (- (/ (* (- t x) (- y a)) z)) t)
         (if (<= t_2 2e+278) t_2 (fma (- y z) t_1 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 - x)) / (a - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = ((y - z) * t_1) + x;
	} else if (t_2 <= -1e-290) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = -(((t - x) * (y - a)) / z) + t;
	} else if (t_2 <= 2e+278) {
		tmp = t_2;
	} else {
		tmp = fma((y - z), t_1, 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(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = Float64(Float64(Float64(y - z) * t_1) + x);
	elseif (t_2 <= -1e-290)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(t - x) * Float64(y - a)) / z)) + t);
	elseif (t_2 <= 2e+278)
		tmp = t_2;
	else
		tmp = fma(Float64(y - z), t_1, 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[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], N[(N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, -1e-290], t$95$2, If[LessEqual[t$95$2, 0.0], N[((-N[(N[(N[(t - x), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision], If[LessEqual[t$95$2, 2e+278], t$95$2, N[(N[(y - z), $MachinePrecision] * t$95$1 + x), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.0000000000000001e-290 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.99999999999999993e278

   1. Initial program 68.1%

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

   if -1.0000000000000001e-290 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   if 1.99999999999999993e278 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

Alternative 3: 91.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ t_2 := x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-290}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\left(-\frac{\left(t - x\right) \cdot \left(y - a\right)}{z}\right) + t\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+278}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) (- a z)) x))
        (t_2 (+ x (/ (* (- y z) (- t x)) (- a z)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -1e-290)
       t_2
       (if (<= t_2 0.0)
         (+ (- (/ (* (- t x) (- y a)) z)) t)
         (if (<= t_2 2e+278) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((t - x) / (a - z)), x);
	double t_2 = x + (((y - z) * (t - x)) / (a - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -1e-290) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = -(((t - x) * (y - a)) / z) + t;
	} else if (t_2 <= 2e+278) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(t - x) / Float64(a - z)), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -1e-290)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(t - x) * Float64(y - a)) / z)) + t);
	elseif (t_2 <= 2e+278)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -1e-290], t$95$2, If[LessEqual[t$95$2, 0.0], N[((-N[(N[(N[(t - x), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision], If[LessEqual[t$95$2, 2e+278], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -inf.0 or 1.99999999999999993e278 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z)))

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < -1.0000000000000001e-290 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 1.99999999999999993e278

   1. Initial program 68.1%

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

   if -1.0000000000000001e-290 < (+.f64 x (/.f64 (*.f64 (-.f64 y z) (-.f64 t x)) (-.f64 a z))) < 0.0

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

Alternative 4: 83.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \left(\left(1 + \left(\frac{a}{z} + \frac{x \cdot \left(y - a\right)}{t \cdot z}\right)\right) - \frac{y}{z}\right)\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{+262}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+208}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t - x}{a - z} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (- (+ 1.0 (+ (/ a z) (/ (* x (- y a)) (* t z)))) (/ y z)))))
   (if (<= z -3.8e+262)
     t_1
     (if (<= z 1.05e+208) (+ (* (- y z) (/ (- t x) (- a z))) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((1.0 + ((a / z) + ((x * (y - a)) / (t * z)))) - (y / z));
	double tmp;
	if (z <= -3.8e+262) {
		tmp = t_1;
	} else if (z <= 1.05e+208) {
		tmp = ((y - z) * ((t - x) / (a - z))) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * ((1.0d0 + ((a / z) + ((x * (y - a)) / (t * z)))) - (y / z))
    if (z <= (-3.8d+262)) then
        tmp = t_1
    else if (z <= 1.05d+208) then
        tmp = ((y - z) * ((t - x) / (a - z))) + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((1.0 + ((a / z) + ((x * (y - a)) / (t * z)))) - (y / z));
	double tmp;
	if (z <= -3.8e+262) {
		tmp = t_1;
	} else if (z <= 1.05e+208) {
		tmp = ((y - z) * ((t - x) / (a - z))) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t * ((1.0 + ((a / z) + ((x * (y - a)) / (t * z)))) - (y / z))
	tmp = 0
	if z <= -3.8e+262:
		tmp = t_1
	elif z <= 1.05e+208:
		tmp = ((y - z) * ((t - x) / (a - z))) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(Float64(1.0 + Float64(Float64(a / z) + Float64(Float64(x * Float64(y - a)) / Float64(t * z)))) - Float64(y / z)))
	tmp = 0.0
	if (z <= -3.8e+262)
		tmp = t_1;
	elseif (z <= 1.05e+208)
		tmp = Float64(Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t * ((1.0 + ((a / z) + ((x * (y - a)) / (t * z)))) - (y / z));
	tmp = 0.0;
	if (z <= -3.8e+262)
		tmp = t_1;
	elseif (z <= 1.05e+208)
		tmp = ((y - z) * ((t - x) / (a - z))) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(1.0 + N[(N[(a / z), $MachinePrecision] + N[(N[(x * N[(y - a), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e+262], t$95$1, If[LessEqual[z, 1.05e+208], N[(N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.80000000000000034e262 or 1.0499999999999999e208 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 44.2

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

   7. Applied rewrites 44.2%

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

   if -3.80000000000000034e262 < z < 1.0499999999999999e208

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

Alternative 5: 80.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 3.2 \cdot 10^{+187}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 3.2e+187)
   (fma (- y z) (/ (- t x) (- a z)) x)
   (- t (/ (* y (- t x)) z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 3.2e+187) {
		tmp = fma((y - z), ((t - x) / (a - z)), x);
	} else {
		tmp = t - ((y * (t - x)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 3.2e+187)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / Float64(a - z)), x);
	else
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 3.2e+187], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 3.19999999999999993e187

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   if 3.19999999999999993e187 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

Alternative 6: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{a - z}\\ \mathbf{if}\;a \leq -2.45 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(y - z, t\_1, x\right)\\ \mathbf{elif}\;a \leq 1.08 \cdot 10^{-5}:\\ \;\;\;\;\left(-\frac{\left(t - x\right) \cdot \left(y - a\right)}{z}\right) + t\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot t\_1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ t (- a z))))
   (if (<= a -2.45e-42)
     (fma (- y z) t_1 x)
     (if (<= a 1.08e-5)
       (+ (- (/ (* (- t x) (- y a)) z)) t)
       (+ (* (- y z) t_1) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t / (a - z);
	double tmp;
	if (a <= -2.45e-42) {
		tmp = fma((y - z), t_1, x);
	} else if (a <= 1.08e-5) {
		tmp = -(((t - x) * (y - a)) / z) + t;
	} else {
		tmp = ((y - z) * t_1) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t / Float64(a - z))
	tmp = 0.0
	if (a <= -2.45e-42)
		tmp = fma(Float64(y - z), t_1, x);
	elseif (a <= 1.08e-5)
		tmp = Float64(Float64(-Float64(Float64(Float64(t - x) * Float64(y - a)) / z)) + t);
	else
		tmp = Float64(Float64(Float64(y - z) * t_1) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.45e-42], N[(N[(y - z), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[a, 1.08e-5], N[((-N[(N[(N[(t - x), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]) + t), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.45e-42

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if -2.45e-42 < a < 1.07999999999999999e-5

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   if 1.07999999999999999e-5 < a

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 64.1%

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

Alternative 7: 74.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{a - z}\\ \mathbf{if}\;a \leq -2.45 \cdot 10^{-42}:\\ \;\;\;\;\mathsf{fma}\left(y - z, t\_1, x\right)\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-181}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{\left(t - x\right) \cdot y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot t\_1 + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ t (- a z))))
   (if (<= a -2.45e-42)
     (fma (- y z) t_1 x)
     (if (<= a 3.7e-181)
       (- t (/ (* y (- t x)) z))
       (if (<= a 3.6e-113)
         (+ x (/ (* (- t x) y) (- a z)))
         (+ (* (- y z) t_1) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t / (a - z);
	double tmp;
	if (a <= -2.45e-42) {
		tmp = fma((y - z), t_1, x);
	} else if (a <= 3.7e-181) {
		tmp = t - ((y * (t - x)) / z);
	} else if (a <= 3.6e-113) {
		tmp = x + (((t - x) * y) / (a - z));
	} else {
		tmp = ((y - z) * t_1) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t / Float64(a - z))
	tmp = 0.0
	if (a <= -2.45e-42)
		tmp = fma(Float64(y - z), t_1, x);
	elseif (a <= 3.7e-181)
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	elseif (a <= 3.6e-113)
		tmp = Float64(x + Float64(Float64(Float64(t - x) * y) / Float64(a - z)));
	else
		tmp = Float64(Float64(Float64(y - z) * t_1) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.45e-42], N[(N[(y - z), $MachinePrecision] * t$95$1 + x), $MachinePrecision], If[LessEqual[a, 3.7e-181], N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e-113], N[(x + N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] * t$95$1), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.45e-42

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if -2.45e-42 < a < 3.69999999999999984e-181

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

   if 3.69999999999999984e-181 < a < 3.59999999999999975e-113

   1. Initial program 68.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 55.6

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

   4. Applied rewrites 55.6%

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

   if 3.59999999999999975e-113 < a

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in x around 0

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

   6. Applied rewrites 64.1%

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

Alternative 8: 73.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{if}\;a \leq -2.45 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.7 \cdot 10^{-181}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{elif}\;a \leq 3.6 \cdot 10^{-113}:\\ \;\;\;\;x + \frac{\left(t - x\right) \cdot y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ t (- a z)) x)))
   (if (<= a -2.45e-42)
     t_1
     (if (<= a 3.7e-181)
       (- t (/ (* y (- t x)) z))
       (if (<= a 3.6e-113) (+ x (/ (* (- t x) y) (- a z))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (t / (a - z)), x);
	double tmp;
	if (a <= -2.45e-42) {
		tmp = t_1;
	} else if (a <= 3.7e-181) {
		tmp = t - ((y * (t - x)) / z);
	} else if (a <= 3.6e-113) {
		tmp = x + (((t - x) * y) / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(t / Float64(a - z)), x)
	tmp = 0.0
	if (a <= -2.45e-42)
		tmp = t_1;
	elseif (a <= 3.7e-181)
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	elseif (a <= 3.6e-113)
		tmp = Float64(x + Float64(Float64(Float64(t - x) * y) / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.45e-42], t$95$1, If[LessEqual[a, 3.7e-181], N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 3.6e-113], N[(x + N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.45e-42 or 3.59999999999999975e-113 < a

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if -2.45e-42 < a < 3.69999999999999984e-181

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

   if 3.69999999999999984e-181 < a < 3.59999999999999975e-113

   1. Initial program 68.1%

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

   2. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lift--.f64 55.6

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

   4. Applied rewrites 55.6%

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

Alternative 9: 73.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{if}\;a \leq -2.45 \cdot 10^{-42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.3 \cdot 10^{-115}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ t (- a z)) x)))
   (if (<= a -2.45e-42)
     t_1
     (if (<= a 2.3e-115) (- t (/ (* y (- t x)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (t / (a - z)), x);
	double tmp;
	if (a <= -2.45e-42) {
		tmp = t_1;
	} else if (a <= 2.3e-115) {
		tmp = t - ((y * (t - x)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(t / Float64(a - z)), x)
	tmp = 0.0
	if (a <= -2.45e-42)
		tmp = t_1;
	elseif (a <= 2.3e-115)
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.45e-42], t$95$1, If[LessEqual[a, 2.3e-115], N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.45e-42 or 2.29999999999999985e-115 < a

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. associate-/l* N/A

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

      9. sub-div N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 N/A

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

      12. sub-div N/A

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

      13. lower-/.f64 N/A

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

      14. lift--.f64 N/A

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

      15. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lift--.f64 64.1

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

   6. Applied rewrites 64.1%

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

   if -2.45e-42 < a < 2.29999999999999985e-115

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

Alternative 10: 68.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -4.5 \cdot 10^{+50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.5 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ (- y z) (- a z)))))
   (if (<= z -4.5e+50)
     t_1
     (if (<= z 1.5e-69)
       (fma (- t x) (/ (- y z) a) x)
       (if (<= z 1.8e+187) t_1 (- t (/ (* y (- t x)) z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((y - z) / (a - z));
	double tmp;
	if (z <= -4.5e+50) {
		tmp = t_1;
	} else if (z <= 1.5e-69) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else if (z <= 1.8e+187) {
		tmp = t_1;
	} else {
		tmp = t - ((y * (t - x)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(Float64(y - z) / Float64(a - z)))
	tmp = 0.0
	if (z <= -4.5e+50)
		tmp = t_1;
	elseif (z <= 1.5e-69)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	elseif (z <= 1.8e+187)
		tmp = t_1;
	else
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.5e+50], t$95$1, If[LessEqual[z, 1.5e-69], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.8e+187], t$95$1, N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.50000000000000014e50 or 1.49999999999999995e-69 < z < 1.80000000000000018e187

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   5. 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 51.5

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

   6. Applied rewrites 51.5%

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

   if -4.50000000000000014e50 < z < 1.49999999999999995e-69

   1. Initial program 68.1%

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

   2. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x + \frac{\left(t - x\right) \cdot \left(y - z\right)}{a}} \]
   3. 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 53.8

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

   4. Applied rewrites 53.8%

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

   if 1.80000000000000018e187 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

Alternative 11: 65.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-103}:\\ \;\;\;\;y \cdot \frac{t - x}{a} + x\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ (- y z) (- a z)))))
   (if (<= z -1.8e+46)
     t_1
     (if (<= z 6.4e-103)
       (+ (* y (/ (- t x) a)) x)
       (if (<= z 1.8e+187) t_1 (- t (/ (* y (- t x)) z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((y - z) / (a - z));
	double tmp;
	if (z <= -1.8e+46) {
		tmp = t_1;
	} else if (z <= 6.4e-103) {
		tmp = (y * ((t - x) / a)) + x;
	} else if (z <= 1.8e+187) {
		tmp = t_1;
	} else {
		tmp = t - ((y * (t - x)) / 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) :: t_1
    real(8) :: tmp
    t_1 = t * ((y - z) / (a - z))
    if (z <= (-1.8d+46)) then
        tmp = t_1
    else if (z <= 6.4d-103) then
        tmp = (y * ((t - x) / a)) + x
    else if (z <= 1.8d+187) then
        tmp = t_1
    else
        tmp = t - ((y * (t - x)) / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((y - z) / (a - z));
	double tmp;
	if (z <= -1.8e+46) {
		tmp = t_1;
	} else if (z <= 6.4e-103) {
		tmp = (y * ((t - x) / a)) + x;
	} else if (z <= 1.8e+187) {
		tmp = t_1;
	} else {
		tmp = t - ((y * (t - x)) / z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t * ((y - z) / (a - z))
	tmp = 0
	if z <= -1.8e+46:
		tmp = t_1
	elif z <= 6.4e-103:
		tmp = (y * ((t - x) / a)) + x
	elif z <= 1.8e+187:
		tmp = t_1
	else:
		tmp = t - ((y * (t - x)) / z)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(Float64(y - z) / Float64(a - z)))
	tmp = 0.0
	if (z <= -1.8e+46)
		tmp = t_1;
	elseif (z <= 6.4e-103)
		tmp = Float64(Float64(y * Float64(Float64(t - x) / a)) + x);
	elseif (z <= 1.8e+187)
		tmp = t_1;
	else
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t * ((y - z) / (a - z));
	tmp = 0.0;
	if (z <= -1.8e+46)
		tmp = t_1;
	elseif (z <= 6.4e-103)
		tmp = (y * ((t - x) / a)) + x;
	elseif (z <= 1.8e+187)
		tmp = t_1;
	else
		tmp = t - ((y * (t - x)) / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e+46], t$95$1, If[LessEqual[z, 6.4e-103], N[(N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.8e+187], t$95$1, N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7999999999999999e46 or 6.39999999999999953e-103 < z < 1.80000000000000018e187

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   5. 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 51.5

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

   6. Applied rewrites 51.5%

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

   if -1.7999999999999999e46 < z < 6.39999999999999953e-103

   1. Initial program 68.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   3. 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 48.1

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

   4. Applied rewrites 48.1%

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

      1. lift-fma.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lower-+.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 48.1

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

   6. Applied rewrites 48.1%

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

   if 1.80000000000000018e187 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

Alternative 12: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{y - z}{a - z}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-103}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+187}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t - \frac{y \cdot \left(t - x\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ (- y z) (- a z)))))
   (if (<= z -1.8e+46)
     t_1
     (if (<= z 6.4e-103)
       (fma y (/ (- t x) a) x)
       (if (<= z 1.8e+187) t_1 (- t (/ (* y (- t x)) z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * ((y - z) / (a - z));
	double tmp;
	if (z <= -1.8e+46) {
		tmp = t_1;
	} else if (z <= 6.4e-103) {
		tmp = fma(y, ((t - x) / a), x);
	} else if (z <= 1.8e+187) {
		tmp = t_1;
	} else {
		tmp = t - ((y * (t - x)) / z);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(Float64(y - z) / Float64(a - z)))
	tmp = 0.0
	if (z <= -1.8e+46)
		tmp = t_1;
	elseif (z <= 6.4e-103)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	elseif (z <= 1.8e+187)
		tmp = t_1;
	else
		tmp = Float64(t - Float64(Float64(y * Float64(t - x)) / z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e+46], t$95$1, If[LessEqual[z, 6.4e-103], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.8e+187], t$95$1, N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.7999999999999999e46 or 6.39999999999999953e-103 < z < 1.80000000000000018e187

   1. Initial program 68.1%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift--.f64 N/A

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

      7. +-commutative N/A

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

      8. lower-+.f64 N/A

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

      9. associate-/l* N/A

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

      10. sub-div N/A

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

      11. lower-*.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-div N/A

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

      14. lower-/.f64 N/A

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

      15. lift--.f64 N/A

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

      16. lift--.f64 80.1

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

   3. Applied rewrites 80.1%

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

   4. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t \cdot \left(\frac{y}{a - z} - \frac{z}{a - z}\right)} \]
   5. 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 51.5

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

   6. Applied rewrites 51.5%

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

   if -1.7999999999999999e46 < z < 6.39999999999999953e-103

   1. Initial program 68.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   3. 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 48.1

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

   4. Applied rewrites 48.1%

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

   if 1.80000000000000018e187 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

Alternative 13: 65.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y \cdot \left(t - x\right)}{z}\\ \mathbf{if}\;z \leq -1.12 \cdot 10^{+67}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 0.00182:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (/ (* y (- t x)) z))))
   (if (<= z -1.12e+67) t_1 (if (<= z 0.00182) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y * (t - x)) / z);
	double tmp;
	if (z <= -1.12e+67) {
		tmp = t_1;
	} else if (z <= 0.00182) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y * Float64(t - x)) / z))
	tmp = 0.0
	if (z <= -1.12e+67)
		tmp = t_1;
	elseif (z <= 0.00182)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.12e+67], t$95$1, If[LessEqual[z, 0.00182], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.12e67 or 0.00182 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in z around -inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. lower--.f64 46.2

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

   4. Applied rewrites 46.2%

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

   5. Taylor expanded in a around 0

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

      1. lower--.f64 N/A

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

      2. lower-/.f64 N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 44.2

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

   7. Applied rewrites 44.2%

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

   if -1.12e67 < z < 0.00182

   1. Initial program 68.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   3. 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 48.1

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

   4. Applied rewrites 48.1%

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

Alternative 14: 59.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+53}:\\ \;\;\;\;t \cdot 1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.65e+53)
   (* t 1.0)
   (if (<= z 8.5e+42) (fma y (/ (- t x) a) x) (* t 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.65e+53) {
		tmp = t * 1.0;
	} else if (z <= 8.5e+42) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.65e+53)
		tmp = Float64(t * 1.0);
	elseif (z <= 8.5e+42)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = Float64(t * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.65e+53], N[(t * 1.0), $MachinePrecision], If[LessEqual[z, 8.5e+42], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(t * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6500000000000001e53 or 8.5000000000000003e42 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto t \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.2%

       \[\leadsto t \cdot 1 \]

   if -1.6500000000000001e53 < z < 8.5000000000000003e42

   1. Initial program 68.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   3. 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 48.1

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

   4. Applied rewrites 48.1%

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

Alternative 15: 52.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.6 \cdot 10^{+53}:\\ \;\;\;\;t \cdot 1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+42}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.6e+53)
   (* t 1.0)
   (if (<= z 8.5e+42) (fma y (/ t a) x) (* t 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.6e+53) {
		tmp = t * 1.0;
	} else if (z <= 8.5e+42) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.6e+53)
		tmp = Float64(t * 1.0);
	elseif (z <= 8.5e+42)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(t * 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.6e+53], N[(t * 1.0), $MachinePrecision], If[LessEqual[z, 8.5e+42], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(t * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.6e53 or 8.5000000000000003e42 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto t \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.2%

       \[\leadsto t \cdot 1 \]

   if -1.6e53 < z < 8.5000000000000003e42

   1. Initial program 68.1%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x + \frac{y \cdot \left(t - x\right)}{a}} \]
   3. 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 48.1

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

   4. Applied rewrites 48.1%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 39.9

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

   7. Applied rewrites 39.9%

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

Alternative 16: 38.0% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.65 \cdot 10^{+36}:\\ \;\;\;\;t \cdot 1\\ \mathbf{elif}\;z \leq 1.3 \cdot 10^{-69}:\\ \;\;\;\;1 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.65e+36) (* t 1.0) (if (<= z 1.3e-69) (* 1.0 x) (* t 1.0))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e+36) {
		tmp = t * 1.0;
	} else if (z <= 1.3e-69) {
		tmp = 1.0 * x;
	} else {
		tmp = t * 1.0;
	}
	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 (z <= (-2.65d+36)) then
        tmp = t * 1.0d0
    else if (z <= 1.3d-69) then
        tmp = 1.0d0 * x
    else
        tmp = t * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.65e+36) {
		tmp = t * 1.0;
	} else if (z <= 1.3e-69) {
		tmp = 1.0 * x;
	} else {
		tmp = t * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.65e+36:
		tmp = t * 1.0
	elif z <= 1.3e-69:
		tmp = 1.0 * x
	else:
		tmp = t * 1.0
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.65e+36)
		tmp = Float64(t * 1.0);
	elseif (z <= 1.3e-69)
		tmp = Float64(1.0 * x);
	else
		tmp = Float64(t * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.65e+36)
		tmp = t * 1.0;
	elseif (z <= 1.3e-69)
		tmp = 1.0 * x;
	else
		tmp = t * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.65e+36], N[(t * 1.0), $MachinePrecision], If[LessEqual[z, 1.3e-69], N[(1.0 * x), $MachinePrecision], N[(t * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.65e36 or 1.3000000000000001e-69 < z

   1. Initial program 68.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto t \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.2%

       \[\leadsto t \cdot 1 \]

   if -2.65e36 < z < 1.3000000000000001e-69

   1. Initial program 68.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in a around inf

      \[\leadsto 1 \cdot x \]
   6. Step-by-step derivation

   7. Applied rewrites 25.3%

      \[\leadsto 1 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 36.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.62 \cdot 10^{+16}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-167}:\\ \;\;\;\;t \cdot 1\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.62e+16) (+ x t) (if (<= a 1.4e-167) (* t 1.0) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.62e+16) {
		tmp = x + t;
	} else if (a <= 1.4e-167) {
		tmp = t * 1.0;
	} else {
		tmp = x + t;
	}
	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 <= (-1.62d+16)) then
        tmp = x + t
    else if (a <= 1.4d-167) then
        tmp = t * 1.0d0
    else
        tmp = x + t
    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 <= -1.62e+16) {
		tmp = x + t;
	} else if (a <= 1.4e-167) {
		tmp = t * 1.0;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1.62e+16:
		tmp = x + t
	elif a <= 1.4e-167:
		tmp = t * 1.0
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.62e+16)
		tmp = Float64(x + t);
	elseif (a <= 1.4e-167)
		tmp = Float64(t * 1.0);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1.62e+16)
		tmp = x + t;
	elseif (a <= 1.4e-167)
		tmp = t * 1.0;
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.62e+16], N[(x + t), $MachinePrecision], If[LessEqual[a, 1.4e-167], N[(t * 1.0), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.62e16 or 1.39999999999999993e-167 < a

   1. Initial program 68.1%

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

   2. Taylor expanded in z around inf

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

      1. lift--.f64 19.3

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

   4. Applied rewrites 19.3%

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

   5. Taylor expanded in x around 0

      \[\leadsto x + t \]
   6. Step-by-step derivation

   7. Applied rewrites 34.1%

      \[\leadsto x + t \]

   if -1.62e16 < a < 1.39999999999999993e-167

   1. Initial program 68.1%

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

   2. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

   4. Applied rewrites 69.1%

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

   5. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 42.4

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

   7. Applied rewrites 42.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto t \cdot 1 \]
   9. Step-by-step derivation

   10. Applied rewrites 25.2%

       \[\leadsto t \cdot 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 18: 34.1% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ x + t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return x + 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 = x + t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + t), $MachinePrecision]

Derivation

1. Initial program 68.1%

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

2. Taylor expanded in z around inf

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

   1. lift--.f64 19.3

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

4. Applied rewrites 19.3%

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

5. Taylor expanded in x around 0

   \[\leadsto x + t \]
6. Step-by-step derivation

7. Applied rewrites 34.1%

   \[\leadsto x + t \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025139 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64
  (+ x (/ (* (- y z) (- t x)) (- a z))))