Development.Shake.Progress:decay from shake-0.15.5

Percentage Accurate: 66.6% → 94.8%

Time: 5.6s

Alternatives: 16

Speedup: 1.2×

Specification

Math

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

FPCore

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

C

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

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, b)
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), intent (in) :: b
    code = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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, b)
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), intent (in) :: b
    code = ((x * y) + (z * (t - a))) / (y + (z * (b - y)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.8% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \mathsf{fma}\left(b - y, z, y\right)\\ t_4 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_5 := \mathsf{fma}\left(z, \frac{t - a}{t\_3}, x \cdot \frac{y}{t\_3}\right)\\ \mathbf{if}\;t\_4 \leq -\infty:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_4 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, \mathsf{fma}\left(t, z, y \cdot x\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_4 \leq 0:\\ \;\;\;\;\left(-\frac{\left(-x \cdot \frac{y}{b - y}\right) - \left(-\frac{\left(t - a\right) \cdot y}{\left(b - y\right) \cdot \left(b - y\right)}\right)}{z}\right) + t\_2\\ \mathbf{elif}\;t\_4 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_4 \leq \infty:\\ \;\;\;\;t\_5\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (fma (- b y) z y))
        (t_4 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_5 (fma z (/ (- t a) t_3) (* x (/ y t_3)))))
   (if (<= t_4 (- INFINITY))
     t_5
     (if (<= t_4 -1e-276)
       (/ (fma (- a) z (fma t z (* y x))) t_1)
       (if (<= t_4 0.0)
         (+
          (-
           (/
            (-
             (- (* x (/ y (- b y))))
             (- (/ (* (- t a) y) (* (- b y) (- b y)))))
            z))
          t_2)
         (if (<= t_4 2e+304) t_4 (if (<= t_4 INFINITY) t_5 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = (t - a) / (b - y);
	double t_3 = fma((b - y), z, y);
	double t_4 = ((x * y) + (z * (t - a))) / t_1;
	double t_5 = fma(z, ((t - a) / t_3), (x * (y / t_3)));
	double tmp;
	if (t_4 <= -((double) INFINITY)) {
		tmp = t_5;
	} else if (t_4 <= -1e-276) {
		tmp = fma(-a, z, fma(t, z, (y * x))) / t_1;
	} else if (t_4 <= 0.0) {
		tmp = -((-(x * (y / (b - y))) - -(((t - a) * y) / ((b - y) * (b - y)))) / z) + t_2;
	} else if (t_4 <= 2e+304) {
		tmp = t_4;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_5;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = fma(Float64(b - y), z, y)
	t_4 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_5 = fma(z, Float64(Float64(t - a) / t_3), Float64(x * Float64(y / t_3)))
	tmp = 0.0
	if (t_4 <= Float64(-Inf))
		tmp = t_5;
	elseif (t_4 <= -1e-276)
		tmp = Float64(fma(Float64(-a), z, fma(t, z, Float64(y * x))) / t_1);
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(-Float64(Float64(Float64(-Float64(x * Float64(y / Float64(b - y)))) - Float64(-Float64(Float64(Float64(t - a) * y) / Float64(Float64(b - y) * Float64(b - y))))) / z)) + t_2);
	elseif (t_4 <= 2e+304)
		tmp = t_4;
	elseif (t_4 <= Inf)
		tmp = t_5;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$4 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$5 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$3), $MachinePrecision] + N[(x * N[(y / t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, (-Infinity)], t$95$5, If[LessEqual[t$95$4, -1e-276], N[(N[((-a) * z + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$4, 0.0], N[((-N[(N[((-N[(x * N[(y / N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]) - (-N[(N[(N[(t - a), $MachinePrecision] * y), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision])), $MachinePrecision] / z), $MachinePrecision]) + t$95$2), $MachinePrecision], If[LessEqual[t$95$4, 2e+304], t$95$4, If[LessEqual[t$95$4, Infinity], t$95$5, t$95$2]]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1e-276

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -1e-276 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0

   1. Initial program 30.2%

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

   2. Taylor expanded in z around -inf

      \[\leadsto \color{blue}{\left(-1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \frac{t}{b - y}\right) - \frac{a}{b - y}} \]
   3. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \color{blue}{\left(\frac{t}{b - y} - \frac{a}{b - y}\right)} \]

      2. div-sub N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \frac{t - a}{\color{blue}{b - y}} \]

      3. lower-+.f64 N/A

         \[\leadsto -1 \cdot \frac{-1 \cdot \frac{x \cdot y}{b - y} - -1 \cdot \frac{y \cdot \left(t - a\right)}{{\left(b - y\right)}^{2}}}{z} + \color{blue}{\frac{t - a}{b - y}} \]

   4. Applied rewrites 83.2%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 99.5%

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

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

   1. Initial program 0.0%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 74.6

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

   4. Applied rewrites 74.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 5 regimes into one program.
4. Add Preprocessing

Alternative 2: 94.2% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \mathsf{fma}\left(b - y, z, y\right)\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{t\_2}, x \cdot \frac{y}{t\_2}\right)\\ t_5 := \frac{t - a}{b - y}\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, \mathsf{fma}\left(t, z, y \cdot x\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_5\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (fma (- b y) z y))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma z (/ (- t a) t_2) (* x (/ y t_2))))
        (t_5 (/ (- t a) (- b y))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-276)
       (/ (fma (- a) z (fma t z (* y x))) t_1)
       (if (<= t_3 0.0)
         t_5
         (if (<= t_3 2e+304) t_3 (if (<= t_3 INFINITY) t_4 t_5)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = fma((b - y), z, y);
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma(z, ((t - a) / t_2), (x * (y / t_2)));
	double t_5 = (t - a) / (b - y);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-276) {
		tmp = fma(-a, z, fma(t, z, (y * x))) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_5;
	} else if (t_3 <= 2e+304) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_5;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = fma(Float64(b - y), z, y)
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(z, Float64(Float64(t - a) / t_2), Float64(x * Float64(y / t_2)))
	t_5 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-276)
		tmp = Float64(fma(Float64(-a), z, fma(t, z, Float64(y * x))) / t_1);
	elseif (t_3 <= 0.0)
		tmp = t_5;
	elseif (t_3 <= 2e+304)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_5;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / t$95$2), $MachinePrecision] + N[(x * N[(y / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-276], N[(N[((-a) * z + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$5, If[LessEqual[t$95$3, 2e+304], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$5]]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1e-276

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -1e-276 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 12.2%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 75.6

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

   4. Applied rewrites 75.6%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 99.5%

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

Alternative 3: 93.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x \cdot \frac{-1}{z - 1}\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, \mathsf{fma}\left(t, z, y \cdot x\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma z (/ (- t a) (fma (- b y) z y)) (* x (/ -1.0 (- z 1.0))))))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-276)
       (/ (fma (- a) z (fma t z (* y x))) t_1)
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 2e+304) t_3 (if (<= t_3 INFINITY) t_4 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = (t - a) / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma(z, ((t - a) / fma((b - y), z, y)), (x * (-1.0 / (z - 1.0))));
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-276) {
		tmp = fma(-a, z, fma(t, z, (y * x))) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 2e+304) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), Float64(x * Float64(-1.0 / Float64(z - 1.0))))
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-276)
		tmp = Float64(fma(Float64(-a), z, fma(t, z, Float64(y * x))) / t_1);
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 2e+304)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + N[(x * N[(-1.0 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-276], N[(N[((-a) * z + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 2e+304], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in y around -inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.9

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

   6. Applied rewrites 95.9%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1e-276

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -1e-276 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 12.2%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 75.6

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

   4. Applied rewrites 75.6%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 99.5%

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

Alternative 4: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y + z \cdot \left(b - y\right)\\ t_2 := \frac{t - a}{b - y}\\ t_3 := \frac{x \cdot y + z \cdot \left(t - a\right)}{t\_1}\\ t_4 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \mathbf{if}\;t\_3 \leq -\infty:\\ \;\;\;\;t\_4\\ \mathbf{elif}\;t\_3 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, \mathsf{fma}\left(t, z, y \cdot x\right)\right)}{t\_1}\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_4\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (+ y (* z (- b y))))
        (t_2 (/ (- t a) (- b y)))
        (t_3 (/ (+ (* x y) (* z (- t a))) t_1))
        (t_4 (fma z (/ (- t a) (fma (- b y) z y)) x)))
   (if (<= t_3 (- INFINITY))
     t_4
     (if (<= t_3 -1e-276)
       (/ (fma (- a) z (fma t z (* y x))) t_1)
       (if (<= t_3 0.0)
         t_2
         (if (<= t_3 2e+304) t_3 (if (<= t_3 INFINITY) t_4 t_2)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = y + (z * (b - y));
	double t_2 = (t - a) / (b - y);
	double t_3 = ((x * y) + (z * (t - a))) / t_1;
	double t_4 = fma(z, ((t - a) / fma((b - y), z, y)), x);
	double tmp;
	if (t_3 <= -((double) INFINITY)) {
		tmp = t_4;
	} else if (t_3 <= -1e-276) {
		tmp = fma(-a, z, fma(t, z, (y * x))) / t_1;
	} else if (t_3 <= 0.0) {
		tmp = t_2;
	} else if (t_3 <= 2e+304) {
		tmp = t_3;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_4;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(y + Float64(z * Float64(b - y)))
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	t_3 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / t_1)
	t_4 = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x)
	tmp = 0.0
	if (t_3 <= Float64(-Inf))
		tmp = t_4;
	elseif (t_3 <= -1e-276)
		tmp = Float64(fma(Float64(-a), z, fma(t, z, Float64(y * x))) / t_1);
	elseif (t_3 <= 0.0)
		tmp = t_2;
	elseif (t_3 <= 2e+304)
		tmp = t_3;
	elseif (t_3 <= Inf)
		tmp = t_4;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]}, Block[{t$95$4 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$3, (-Infinity)], t$95$4, If[LessEqual[t$95$3, -1e-276], N[(N[((-a) * z + N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[t$95$3, 0.0], t$95$2, If[LessEqual[t$95$3, 2e+304], t$95$3, If[LessEqual[t$95$3, Infinity], t$95$4, t$95$2]]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 83.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1e-276

   1. Initial program 99.6%

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

   2. Taylor expanded in t around 0

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

      1. associate-*r* N/A

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

      2. mul-1-neg N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 99.6

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

   4. Applied rewrites 99.6%

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

   if -1e-276 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 12.2%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 75.6

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

   4. Applied rewrites 75.6%

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

   if 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 99.5%

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

Alternative 5: 91.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ t_2 := \frac{x \cdot y + z \cdot \left(t - a\right)}{y + z \cdot \left(b - y\right)}\\ t_3 := \mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-276}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+304}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y)))
        (t_2 (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))
        (t_3 (fma z (/ (- t a) (fma (- b y) z y)) x)))
   (if (<= t_2 (- INFINITY))
     t_3
     (if (<= t_2 -1e-276)
       t_2
       (if (<= t_2 0.0)
         t_1
         (if (<= t_2 2e+304) t_2 (if (<= t_2 INFINITY) t_3 t_1)))))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double t_2 = ((x * y) + (z * (t - a))) / (y + (z * (b - y)));
	double t_3 = fma(z, ((t - a) / fma((b - y), z, y)), x);
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_3;
	} else if (t_2 <= -1e-276) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = t_1;
	} else if (t_2 <= 2e+304) {
		tmp = t_2;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	t_2 = Float64(Float64(Float64(x * y) + Float64(z * Float64(t - a))) / Float64(y + Float64(z * Float64(b - y))))
	t_3 = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x)
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_3;
	elseif (t_2 <= -1e-276)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = t_1;
	elseif (t_2 <= 2e+304)
		tmp = t_2;
	elseif (t_2 <= Inf)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(x * y), $MachinePrecision] + N[(z * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y + N[(z * N[(b - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$3, If[LessEqual[t$95$2, -1e-276], t$95$2, If[LessEqual[t$95$2, 0.0], t$95$1, If[LessEqual[t$95$2, 2e+304], t$95$2, If[LessEqual[t$95$2, Infinity], t$95$3, t$95$1]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -inf.0 or 1.9999999999999999e304 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < +inf.0

   1. Initial program 28.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 99.6%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 83.2%

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

   if -inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < -1e-276 or 0.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 1.9999999999999999e304

   1. Initial program 99.5%

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

   if -1e-276 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y)))) < 0.0 or +inf.0 < (/.f64 (+.f64 (*.f64 x y) (*.f64 z (-.f64 t a))) (+.f64 y (*.f64 z (-.f64 b y))))

   1. Initial program 12.2%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 75.6

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

   4. Applied rewrites 75.6%

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -0.0034:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.05 \cdot 10^{-93}:\\ \;\;\;\;\frac{\mathsf{fma}\left(-a, z, y \cdot x\right)}{t\_1}\\ \mathbf{elif}\;z \leq 0.0027:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{t\_1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -0.0034)
     t_2
     (if (<= z -1.05e-93)
       (/ (fma (- a) z (* y x)) t_1)
       (if (<= z 0.0027) (fma z (/ (- t a) t_1) x) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -0.0034) {
		tmp = t_2;
	} else if (z <= -1.05e-93) {
		tmp = fma(-a, z, (y * x)) / t_1;
	} else if (z <= 0.0027) {
		tmp = fma(z, ((t - a) / t_1), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -0.0034)
		tmp = t_2;
	elseif (z <= -1.05e-93)
		tmp = Float64(fma(Float64(-a), z, Float64(y * x)) / t_1);
	elseif (z <= 0.0027)
		tmp = fma(z, Float64(Float64(t - a) / t_1), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -0.0034], t$95$2, If[LessEqual[z, -1.05e-93], N[(N[((-a) * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 0.0027], N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -0.00339999999999999981 or 0.0027000000000000001 < z

   1. Initial program 45.8%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 79.2

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

   4. Applied rewrites 79.2%

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

   if -0.00339999999999999981 < z < -1.05e-93

   1. Initial program 86.0%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. associate-*r* N/A

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

      3. mul-1-neg N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-neg.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lift--.f64 61.5

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

   4. Applied rewrites 61.5%

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

   if -1.05e-93 < z < 0.0027000000000000001

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 89.2%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 74.9%

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

Alternative 7: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(b - y, z, y\right)\\ t_2 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.15 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -3.4 \cdot 10^{-258}:\\ \;\;\;\;\frac{\mathsf{fma}\left(t, z, y \cdot x\right)}{t\_1}\\ \mathbf{elif}\;z \leq 0.0027:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{t\_1}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (fma (- b y) z y)) (t_2 (/ (- t a) (- b y))))
   (if (<= z -1.15e-8)
     t_2
     (if (<= z -3.4e-258)
       (/ (fma t z (* y x)) t_1)
       (if (<= z 0.0027) (fma z (/ (- t a) t_1) x) t_2)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = fma((b - y), z, y);
	double t_2 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.15e-8) {
		tmp = t_2;
	} else if (z <= -3.4e-258) {
		tmp = fma(t, z, (y * x)) / t_1;
	} else if (z <= 0.0027) {
		tmp = fma(z, ((t - a) / t_1), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = fma(Float64(b - y), z, y)
	t_2 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.15e-8)
		tmp = t_2;
	elseif (z <= -3.4e-258)
		tmp = Float64(fma(t, z, Float64(y * x)) / t_1);
	elseif (z <= 0.0027)
		tmp = fma(z, Float64(Float64(t - a) / t_1), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.15e-8], t$95$2, If[LessEqual[z, -3.4e-258], N[(N[(t * z + N[(y * x), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision], If[LessEqual[z, 0.0027], N[(z * N[(N[(t - a), $MachinePrecision] / t$95$1), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.15e-8 or 0.0027000000000000001 < z

   1. Initial program 46.2%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 78.9

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

   4. Applied rewrites 78.9%

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

   if -1.15e-8 < z < -3.3999999999999998e-258

   1. Initial program 86.6%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. +-commutative N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lift--.f64 64.7

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

   4. Applied rewrites 64.7%

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

   if -3.3999999999999998e-258 < z < 0.0027000000000000001

   1. Initial program 87.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 89.0%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 74.3%

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

Alternative 8: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -160000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(t + \frac{x \cdot y}{z}\right) - a}{b}\\ \mathbf{elif}\;z \leq 0.0027:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{t - a}{\mathsf{fma}\left(b - y, z, y\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -160000.0)
     t_1
     (if (<= z -3.8e-78)
       (/ (- (+ t (/ (* x y) z)) a) b)
       (if (<= z 0.0027) (fma z (/ (- t a) (fma (- b y) z y)) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -160000.0) {
		tmp = t_1;
	} else if (z <= -3.8e-78) {
		tmp = ((t + ((x * y) / z)) - a) / b;
	} else if (z <= 0.0027) {
		tmp = fma(z, ((t - a) / fma((b - y), z, y)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -160000.0)
		tmp = t_1;
	elseif (z <= -3.8e-78)
		tmp = Float64(Float64(Float64(t + Float64(Float64(x * y) / z)) - a) / b);
	elseif (z <= 0.0027)
		tmp = fma(z, Float64(Float64(t - a) / fma(Float64(b - y), z, y)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -160000.0], t$95$1, If[LessEqual[z, -3.8e-78], N[(N[(N[(t + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 0.0027], N[(z * N[(N[(t - a), $MachinePrecision] / N[(N[(b - y), $MachinePrecision] * z + y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e5 or 0.0027000000000000001 < z

   1. Initial program 45.3%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 79.6

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

   4. Applied rewrites 79.6%

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

   if -1.6e5 < z < -3.7999999999999999e-78

   1. Initial program 86.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) + \frac{t}{b}\right) - \frac{a}{b}} \]
   3. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto y \cdot \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) + \color{blue}{\left(\frac{t}{b} - \frac{a}{b}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) \cdot y + \left(\color{blue}{\frac{t}{b}} - \frac{a}{b}\right) \]

      3. div-sub N/A

         \[\leadsto \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) \cdot y + \frac{t - a}{\color{blue}{b}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}, \color{blue}{y}, \frac{t - a}{b}\right) \]

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

   if -3.7999999999999999e-78 < z < 0.0027000000000000001

   1. Initial program 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-+.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. lift--.f64 N/A

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

      9. div-add-rev N/A

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

      10. +-commutative N/A

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

      11. associate-/l* N/A

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

      12. lower-fma.f64 N/A

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

   3. Applied rewrites 89.5%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 75.0%

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

Alternative 9: 70.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -160000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-78}:\\ \;\;\;\;\frac{\left(t + \frac{x \cdot y}{z}\right) - a}{b}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{t - a}{y} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -160000.0)
     t_1
     (if (<= z -3.8e-78)
       (/ (- (+ t (/ (* x y) z)) a) b)
       (if (<= z 7.5e-31) (+ (* (/ (- t a) y) z) x) t_1)))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -160000.0) {
		tmp = t_1;
	} else if (z <= -3.8e-78) {
		tmp = ((t + ((x * y) / z)) - a) / b;
	} else if (z <= 7.5e-31) {
		tmp = (((t - a) / y) * 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-160000.0d0)) then
        tmp = t_1
    else if (z <= (-3.8d-78)) then
        tmp = ((t + ((x * y) / z)) - a) / b
    else if (z <= 7.5d-31) then
        tmp = (((t - a) / y) * 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 b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -160000.0) {
		tmp = t_1;
	} else if (z <= -3.8e-78) {
		tmp = ((t + ((x * y) / z)) - a) / b;
	} else if (z <= 7.5e-31) {
		tmp = (((t - a) / y) * z) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -160000.0:
		tmp = t_1
	elif z <= -3.8e-78:
		tmp = ((t + ((x * y) / z)) - a) / b
	elif z <= 7.5e-31:
		tmp = (((t - a) / y) * z) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -160000.0)
		tmp = t_1;
	elseif (z <= -3.8e-78)
		tmp = Float64(Float64(Float64(t + Float64(Float64(x * y) / z)) - a) / b);
	elseif (z <= 7.5e-31)
		tmp = Float64(Float64(Float64(Float64(t - a) / y) * z) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -160000.0)
		tmp = t_1;
	elseif (z <= -3.8e-78)
		tmp = ((t + ((x * y) / z)) - a) / b;
	elseif (z <= 7.5e-31)
		tmp = (((t - a) / y) * z) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -160000.0], t$95$1, If[LessEqual[z, -3.8e-78], N[(N[(N[(t + N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision] - a), $MachinePrecision] / b), $MachinePrecision], If[LessEqual[z, 7.5e-31], N[(N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.6e5 or 7.49999999999999975e-31 < z

   1. Initial program 47.1%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 77.7

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

   4. Applied rewrites 77.7%

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

   if -1.6e5 < z < -3.7999999999999999e-78

   1. Initial program 86.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\left(y \cdot \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) + \frac{t}{b}\right) - \frac{a}{b}} \]
   3. Step-by-step derivation

      1. associate--l+ N/A

         \[\leadsto y \cdot \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) + \color{blue}{\left(\frac{t}{b} - \frac{a}{b}\right)} \]

      2. *-commutative N/A

         \[\leadsto \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) \cdot y + \left(\color{blue}{\frac{t}{b}} - \frac{a}{b}\right) \]

      3. div-sub N/A

         \[\leadsto \left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}\right) \cdot y + \frac{t - a}{\color{blue}{b}} \]

      4. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(\frac{x}{b \cdot z} - \frac{\left(1 + -1 \cdot z\right) \cdot \left(t - a\right)}{{b}^{2} \cdot z}, \color{blue}{y}, \frac{t - a}{b}\right) \]

   4. Applied rewrites 34.7%

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

   5. Taylor expanded in b around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 N/A

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

      3. lower-+.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 39.9

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

   7. Applied rewrites 39.9%

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

   if -3.7999999999999999e-78 < z < 7.49999999999999975e-31

   1. Initial program 87.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 52.1%

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

      1. lift-fma.f64 N/A

         \[\leadsto x \cdot z + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot z + \color{blue}{x} \]

      3. lower-*.f64 52.1

         \[\leadsto x \cdot z + x \]

   9. Applied rewrites 52.1%

      \[\leadsto x \cdot z + \color{blue}{x} \]

   10. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lift--.f64 67.3

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

   12. Applied rewrites 67.3%

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

Alternative 10: 70.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\frac{t - a}{y} \cdot z + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.8e-78) t_1 (if (<= z 7.5e-31) (+ (* (/ (- t a) y) z) x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.8e-78) {
		tmp = t_1;
	} else if (z <= 7.5e-31) {
		tmp = (((t - a) / y) * 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / (b - y)
    if (z <= (-3.8d-78)) then
        tmp = t_1
    else if (z <= 7.5d-31) then
        tmp = (((t - a) / y) * 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 b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.8e-78) {
		tmp = t_1;
	} else if (z <= 7.5e-31) {
		tmp = (((t - a) / y) * z) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / (b - y)
	tmp = 0
	if z <= -3.8e-78:
		tmp = t_1
	elif z <= 7.5e-31:
		tmp = (((t - a) / y) * z) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.8e-78)
		tmp = t_1;
	elseif (z <= 7.5e-31)
		tmp = Float64(Float64(Float64(Float64(t - a) / y) * z) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / (b - y);
	tmp = 0.0;
	if (z <= -3.8e-78)
		tmp = t_1;
	elseif (z <= 7.5e-31)
		tmp = (((t - a) / y) * z) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e-78], t$95$1, If[LessEqual[z, 7.5e-31], N[(N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999999e-78 or 7.49999999999999975e-31 < z

   1. Initial program 51.7%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 72.6

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

   4. Applied rewrites 72.6%

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

   if -3.7999999999999999e-78 < z < 7.49999999999999975e-31

   1. Initial program 87.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma}\left(x, z, x\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 52.1%

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

      1. lift-fma.f64 N/A

         \[\leadsto x \cdot z + \color{blue}{x} \]

      2. lower-+.f64 N/A

         \[\leadsto x \cdot z + \color{blue}{x} \]

      3. lower-*.f64 52.1

         \[\leadsto x \cdot z + x \]

   9. Applied rewrites 52.1%

      \[\leadsto x \cdot z + \color{blue}{x} \]

   10. Taylor expanded in x around 0

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

       1. lower-/.f64 N/A

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

       2. lift--.f64 67.3

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

   12. Applied rewrites 67.3%

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

Alternative 11: 70.4% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -3.8 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - a}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -3.8e-78) t_1 (if (<= z 7.5e-31) (fma (/ (- t a) y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -3.8e-78) {
		tmp = t_1;
	} else if (z <= 7.5e-31) {
		tmp = fma(((t - a) / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -3.8e-78)
		tmp = t_1;
	elseif (z <= 7.5e-31)
		tmp = fma(Float64(Float64(t - a) / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.8e-78], t$95$1, If[LessEqual[z, 7.5e-31], N[(N[(N[(t - a), $MachinePrecision] / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7999999999999999e-78 or 7.49999999999999975e-31 < z

   1. Initial program 51.7%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 72.6

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

   4. Applied rewrites 72.6%

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

   if -3.7999999999999999e-78 < z < 7.49999999999999975e-31

   1. Initial program 87.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 53.3

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

   4. Applied rewrites 53.3%

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

   5. Taylor expanded in x around 0

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

   7. Applied rewrites 67.3%

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

Alternative 12: 66.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b - y}\\ \mathbf{if}\;z \leq -1.65 \cdot 10^{-78}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-90}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{y}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) (- b y))))
   (if (<= z -1.65e-78) t_1 (if (<= z 1.52e-90) (fma (/ t y) z x) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / (b - y);
	double tmp;
	if (z <= -1.65e-78) {
		tmp = t_1;
	} else if (z <= 1.52e-90) {
		tmp = fma((t / y), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / Float64(b - y))
	tmp = 0.0
	if (z <= -1.65e-78)
		tmp = t_1;
	elseif (z <= 1.52e-90)
		tmp = fma(Float64(t / y), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.65e-78], t$95$1, If[LessEqual[z, 1.52e-90], N[(N[(t / y), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.64999999999999991e-78 or 1.51999999999999991e-90 < z

   1. Initial program 54.7%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 69.1

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

   4. Applied rewrites 69.1%

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

   if -1.64999999999999991e-78 < z < 1.51999999999999991e-90

   1. Initial program 87.4%

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

   2. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. div-add-rev N/A

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

      5. sub-div N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 N/A

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

      11. lift--.f64 54.0

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

   4. Applied rewrites 54.0%

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

   5. Taylor expanded in t around inf

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

   7. Applied rewrites 60.7%

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

Alternative 13: 54.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-x}{z - 1}\\ \mathbf{if}\;y \leq -1.45 \cdot 10^{+34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.8 \cdot 10^{-33}:\\ \;\;\;\;\frac{t - a}{b}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- x) (- z 1.0))))
   (if (<= y -1.45e+34) t_1 (if (<= y 2.8e-33) (/ (- t a) b) t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = -x / (z - 1.0);
	double tmp;
	if (y <= -1.45e+34) {
		tmp = t_1;
	} else if (y <= 2.8e-33) {
		tmp = (t - a) / b;
	} 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -x / (z - 1.0d0)
    if (y <= (-1.45d+34)) then
        tmp = t_1
    else if (y <= 2.8d-33) then
        tmp = (t - a) / b
    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 b) {
	double t_1 = -x / (z - 1.0);
	double tmp;
	if (y <= -1.45e+34) {
		tmp = t_1;
	} else if (y <= 2.8e-33) {
		tmp = (t - a) / b;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = -x / (z - 1.0)
	tmp = 0
	if y <= -1.45e+34:
		tmp = t_1
	elif y <= 2.8e-33:
		tmp = (t - a) / b
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(-x) / Float64(z - 1.0))
	tmp = 0.0
	if (y <= -1.45e+34)
		tmp = t_1;
	elseif (y <= 2.8e-33)
		tmp = Float64(Float64(t - a) / b);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = -x / (z - 1.0);
	tmp = 0.0;
	if (y <= -1.45e+34)
		tmp = t_1;
	elseif (y <= 2.8e-33)
		tmp = (t - a) / b;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[((-x) / N[(z - 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.45e+34], t$95$1, If[LessEqual[y, 2.8e-33], N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.4500000000000001e34 or 2.8e-33 < y

   1. Initial program 53.8%

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

   2. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{z - 1}} \]
   3. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{z - 1}} \]

      2. lower-/.f64 N/A

         \[\leadsto \frac{-1 \cdot x}{\color{blue}{z - 1}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{\mathsf{neg}\left(x\right)}{\color{blue}{z} - 1} \]

      4. lower-neg.f64 N/A

         \[\leadsto \frac{-x}{\color{blue}{z} - 1} \]

      5. lower--.f64 52.4

         \[\leadsto \frac{-x}{z - \color{blue}{1}} \]

   4. Applied rewrites 52.4%

      \[\leadsto \color{blue}{\frac{-x}{z - 1}} \]

   if -1.4500000000000001e34 < y < 2.8e-33

   1. Initial program 79.6%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b}} \]

      2. lift--.f64 56.3

         \[\leadsto \frac{t - a}{b} \]

   4. Applied rewrites 56.3%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 47.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - a}{b}\\ \mathbf{if}\;z \leq -1.24 \cdot 10^{-79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ (- t a) b)))
   (if (<= z -1.24e-79) t_1 (if (<= z 1.25e-90) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (z <= -1.24e-79) {
		tmp = t_1;
	} else if (z <= 1.25e-90) {
		tmp = 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - a) / b
    if (z <= (-1.24d-79)) then
        tmp = t_1
    else if (z <= 1.25d-90) then
        tmp = 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 b) {
	double t_1 = (t - a) / b;
	double tmp;
	if (z <= -1.24e-79) {
		tmp = t_1;
	} else if (z <= 1.25e-90) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = (t - a) / b
	tmp = 0
	if z <= -1.24e-79:
		tmp = t_1
	elif z <= 1.25e-90:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(Float64(t - a) / b)
	tmp = 0.0
	if (z <= -1.24e-79)
		tmp = t_1;
	elseif (z <= 1.25e-90)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = (t - a) / b;
	tmp = 0.0;
	if (z <= -1.24e-79)
		tmp = t_1;
	elseif (z <= 1.25e-90)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(N[(t - a), $MachinePrecision] / b), $MachinePrecision]}, If[LessEqual[z, -1.24e-79], t$95$1, If[LessEqual[z, 1.25e-90], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.24000000000000006e-79 or 1.25000000000000005e-90 < z

   1. Initial program 54.7%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{t - a}{\color{blue}{b}} \]

      2. lift--.f64 44.1

         \[\leadsto \frac{t - a}{b} \]

   4. Applied rewrites 44.1%

      \[\leadsto \color{blue}{\frac{t - a}{b}} \]

   if -1.24000000000000006e-79 < z < 1.25000000000000005e-90

   1. Initial program 87.4%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 54.1%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 43.8% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{b - y}\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-90}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b)
 :precision binary64
 (let* ((t_1 (/ t (- b y))))
   (if (<= z -1.8e-72) t_1 (if (<= z 1.52e-90) x t_1))))

C

double code(double x, double y, double z, double t, double a, double b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -1.8e-72) {
		tmp = t_1;
	} else if (z <= 1.52e-90) {
		tmp = 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, b)
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), intent (in) :: b
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (b - y)
    if (z <= (-1.8d-72)) then
        tmp = t_1
    else if (z <= 1.52d-90) then
        tmp = 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 b) {
	double t_1 = t / (b - y);
	double tmp;
	if (z <= -1.8e-72) {
		tmp = t_1;
	} else if (z <= 1.52e-90) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a, b):
	t_1 = t / (b - y)
	tmp = 0
	if z <= -1.8e-72:
		tmp = t_1
	elif z <= 1.52e-90:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a, b)
	t_1 = Float64(t / Float64(b - y))
	tmp = 0.0
	if (z <= -1.8e-72)
		tmp = t_1;
	elseif (z <= 1.52e-90)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a, b)
	t_1 = t / (b - y);
	tmp = 0.0;
	if (z <= -1.8e-72)
		tmp = t_1;
	elseif (z <= 1.52e-90)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := Block[{t$95$1 = N[(t / N[(b - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.8e-72], t$95$1, If[LessEqual[z, 1.52e-90], x, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8e-72 or 1.51999999999999991e-90 < z

   1. Initial program 54.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{t - a}{b - y}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 69.5

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

   4. Applied rewrites 69.5%

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

   5. Taylor expanded in t around inf

      \[\leadsto \frac{t}{\color{blue}{b} - y} \]
   6. Step-by-step derivation

   7. Applied rewrites 38.0%

      \[\leadsto \frac{t}{\color{blue}{b} - y} \]

   if -1.8e-72 < z < 1.51999999999999991e-90

   1. Initial program 87.5%

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

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 53.8%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 26.2% accurate, 23.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a, b)
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), intent (in) :: b
    code = x
end function

Java

public static double code(double x, double y, double z, double t, double a, double b) {
	return x;
}

Python

def code(x, y, z, t, a, b):
	return x

Julia

function code(x, y, z, t, a, b)
	return x
end

MATLAB

function tmp = code(x, y, z, t, a, b)
	tmp = x;
end

Wolfram

code[x_, y_, z_, t_, a_, b_] := x

Derivation

1. Initial program 66.6%

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

2. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x} \]
3. Step-by-step derivation

4. Applied rewrites 26.2%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025112 
(FPCore (x y z t a b)
  :name "Development.Shake.Progress:decay from shake-0.15.5"
  :precision binary64
  (/ (+ (* x y) (* z (- t a))) (+ y (* z (- b y)))))