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

Percentage Accurate: 66.9% → 82.6%

Time: 12.8s

Alternatives: 16

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 66.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 82.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+45} \lor \neg \left(z \leq 2.1 \cdot 10^{+69}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.75e+45) (not (<= z 2.1e+69)))
   (fma (/ (- (- t x)) z) (- y a) t)
   (+ x (/ (* (- y z) (- t x)) (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.75e+45) || !(z <= 2.1e+69)) {
		tmp = fma((-(t - x) / z), (y - a), t);
	} else {
		tmp = x + (((y - z) * (t - x)) / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.75e+45) || !(z <= 2.1e+69))
		tmp = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t);
	else
		tmp = Float64(x + Float64(Float64(Float64(y - z) * Float64(t - x)) / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.75e+45], N[Not[LessEqual[z, 2.1e+69]], $MachinePrecision]], N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], N[(x + N[(N[(N[(y - z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.75e45 or 2.10000000000000015e69 < z

   1. Initial program 33.6%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 79.8%

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

   if -2.75e45 < z < 2.10000000000000015e69

   1. Initial program 88.3%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{+45} \lor \neg \left(z \leq 2.1 \cdot 10^{+69}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 45.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{a}, z, x\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-196}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) a) z x)))
   (if (<= a -2.3e+21)
     t_1
     (if (<= a 1.6e-196)
       (* (/ (- x t) z) y)
       (if (<= a 1.4e-69) t (if (<= a 6.4e+91) (* (/ (- t x) a) y) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / a), z, x);
	double tmp;
	if (a <= -2.3e+21) {
		tmp = t_1;
	} else if (a <= 1.6e-196) {
		tmp = ((x - t) / z) * y;
	} else if (a <= 1.4e-69) {
		tmp = t;
	} else if (a <= 6.4e+91) {
		tmp = ((t - x) / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / a), z, x)
	tmp = 0.0
	if (a <= -2.3e+21)
		tmp = t_1;
	elseif (a <= 1.6e-196)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	elseif (a <= 1.4e-69)
		tmp = t;
	elseif (a <= 6.4e+91)
		tmp = Float64(Float64(Float64(t - x) / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision]}, If[LessEqual[a, -2.3e+21], t$95$1, If[LessEqual[a, 1.6e-196], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[a, 1.4e-69], t, If[LessEqual[a, 6.4e+91], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -2.3e21 or 6.39999999999999979e91 < a

   1. Initial program 66.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 59.8%

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

   if -2.3e21 < a < 1.6e-196

   1. Initial program 63.8%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 53.2%

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

   if 1.6e-196 < a < 1.3999999999999999e-69

   1. Initial program 48.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 9.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 3.1%

       \[\leadsto 0 \cdot x \]

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 58.1%

       \[\leadsto 0 + \color{blue}{t} \]

   if 1.3999999999999999e-69 < a < 6.39999999999999979e91

   1. Initial program 73.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 58.2

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

   8. Applied rewrites 58.2%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 48.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.3 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{a}, z, x\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-196}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 6.4 \cdot 10^{+91}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{a}, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 41.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{z}{a - z}, x\right)\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-196}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma x (/ z (- a z)) x)))
   (if (<= a -5.5e+22)
     t_1
     (if (<= a 1.6e-196)
       (* (/ (- x t) z) y)
       (if (<= a 1.4e-69) t (if (<= a 3.9e+154) (* (/ (- t x) a) y) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(x, (z / (a - z)), x);
	double tmp;
	if (a <= -5.5e+22) {
		tmp = t_1;
	} else if (a <= 1.6e-196) {
		tmp = ((x - t) / z) * y;
	} else if (a <= 1.4e-69) {
		tmp = t;
	} else if (a <= 3.9e+154) {
		tmp = ((t - x) / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(x, Float64(z / Float64(a - z)), x)
	tmp = 0.0
	if (a <= -5.5e+22)
		tmp = t_1;
	elseif (a <= 1.6e-196)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	elseif (a <= 1.4e-69)
		tmp = t;
	elseif (a <= 3.9e+154)
		tmp = Float64(Float64(Float64(t - x) / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.5e+22], t$95$1, If[LessEqual[a, 1.6e-196], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[a, 1.4e-69], t, If[LessEqual[a, 3.9e+154], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.50000000000000021e22 or 3.9000000000000003e154 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.5%

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

   if -5.50000000000000021e22 < a < 1.6e-196

   1. Initial program 63.8%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 53.2%

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

   if 1.6e-196 < a < 1.3999999999999999e-69

   1. Initial program 48.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 9.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 3.1%

       \[\leadsto 0 \cdot x \]

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 58.1%

       \[\leadsto 0 + \color{blue}{t} \]

   if 1.3999999999999999e-69 < a < 3.9000000000000003e154

   1. Initial program 69.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 61.0

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

   8. Applied rewrites 61.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 42.8%

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

4. Final simplification 52.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-196}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a - z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 41.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-196}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma x (/ z a) x)))
   (if (<= a -5.5e+22)
     t_1
     (if (<= a 1.6e-196)
       (* (/ (- x t) z) y)
       (if (<= a 1.4e-69) t (if (<= a 3.9e+154) (* (/ (- t x) a) y) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(x, (z / a), x);
	double tmp;
	if (a <= -5.5e+22) {
		tmp = t_1;
	} else if (a <= 1.6e-196) {
		tmp = ((x - t) / z) * y;
	} else if (a <= 1.4e-69) {
		tmp = t;
	} else if (a <= 3.9e+154) {
		tmp = ((t - x) / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(x, Float64(z / a), x)
	tmp = 0.0
	if (a <= -5.5e+22)
		tmp = t_1;
	elseif (a <= 1.6e-196)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	elseif (a <= 1.4e-69)
		tmp = t;
	elseif (a <= 3.9e+154)
		tmp = Float64(Float64(Float64(t - x) / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -5.5e+22], t$95$1, If[LessEqual[a, 1.6e-196], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[a, 1.4e-69], t, If[LessEqual[a, 3.9e+154], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -5.50000000000000021e22 or 3.9000000000000003e154 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 70.1

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

   5. Applied rewrites 70.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.5%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 52.8%

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

   if -5.50000000000000021e22 < a < 1.6e-196

   1. Initial program 63.8%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 53.2%

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

   if 1.6e-196 < a < 1.3999999999999999e-69

   1. Initial program 48.8%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 9.2%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 3.1%

       \[\leadsto 0 \cdot x \]

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 58.1%

       \[\leadsto 0 + \color{blue}{t} \]

   if 1.3999999999999999e-69 < a < 3.9000000000000003e154

   1. Initial program 69.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 57.3

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 61.0

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

   8. Applied rewrites 61.0%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 42.8%

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

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -5.5 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.6 \cdot 10^{-196}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;a \leq 1.4 \cdot 10^{-69}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+154}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3700 \lor \neg \left(z \leq 2.05 \cdot 10^{+65}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -3700.0) (not (<= z 2.05e+65)))
   (fma (/ (- (- t x)) z) (- y a) t)
   (fma (- t x) (/ (- y z) a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -3700.0) || !(z <= 2.05e+65)) {
		tmp = fma((-(t - x) / z), (y - a), t);
	} else {
		tmp = fma((t - x), ((y - z) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -3700.0) || !(z <= 2.05e+65))
		tmp = fma(Float64(Float64(-Float64(t - x)) / z), Float64(y - a), t);
	else
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -3700.0], N[Not[LessEqual[z, 2.05e+65]], $MachinePrecision]], N[(N[((-N[(t - x), $MachinePrecision]) / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3700 or 2.0500000000000001e65 < z

   1. Initial program 36.3%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 79.2%

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

   if -3700 < z < 2.0500000000000001e65

   1. Initial program 88.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 82.0

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

   8. Applied rewrites 82.0%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3700 \lor \neg \left(z \leq 2.05 \cdot 10^{+65}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{-\left(t - x\right)}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 39.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -3.7 \cdot 10^{+45}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.62 \cdot 10^{-52}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{+154}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma x (/ z a) x)))
   (if (<= a -3.7e+45)
     t_1
     (if (<= a 1.62e-52)
       (fma a (/ t z) t)
       (if (<= a 3.9e+154) (* t (/ y (- a z))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(x, (z / a), x);
	double tmp;
	if (a <= -3.7e+45) {
		tmp = t_1;
	} else if (a <= 1.62e-52) {
		tmp = fma(a, (t / z), t);
	} else if (a <= 3.9e+154) {
		tmp = t * (y / (a - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(x, Float64(z / a), x)
	tmp = 0.0
	if (a <= -3.7e+45)
		tmp = t_1;
	elseif (a <= 1.62e-52)
		tmp = fma(a, Float64(t / z), t);
	elseif (a <= 3.9e+154)
		tmp = Float64(t * Float64(y / Float64(a - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3.7e+45], t$95$1, If[LessEqual[a, 1.62e-52], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[a, 3.9e+154], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -3.69999999999999977e45 or 3.9000000000000003e154 < a

   1. Initial program 67.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 71.7

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

   5. Applied rewrites 71.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 55.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 54.5%

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

   if -3.69999999999999977e45 < a < 1.61999999999999995e-52

   1. Initial program 62.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 30.6

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

   5. Applied rewrites 30.6%

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

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 38.2%

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

   if 1.61999999999999995e-52 < a < 3.9000000000000003e154

   1. Initial program 69.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 58.7

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

   5. Applied rewrites 58.7%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 38.0%

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

Alternative 7: 42.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -225:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{elif}\;z \leq 82000:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -225.0)
   (fma a (/ t z) t)
   (if (<= z 1.1e-165)
     (* (/ (- t x) a) y)
     (if (<= z 82000.0) (fma x (/ z a) x) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -225.0) {
		tmp = fma(a, (t / z), t);
	} else if (z <= 1.1e-165) {
		tmp = ((t - x) / a) * y;
	} else if (z <= 82000.0) {
		tmp = fma(x, (z / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -225.0)
		tmp = fma(a, Float64(t / z), t);
	elseif (z <= 1.1e-165)
		tmp = Float64(Float64(Float64(t - x) / a) * y);
	elseif (z <= 82000.0)
		tmp = fma(x, Float64(z / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -225.0], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.1e-165], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 82000.0], N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t]]]

Derivation

1. Split input into 4 regimes

2. if z < -225

   1. Initial program 44.6%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 48.1

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

   5. Applied rewrites 48.1%

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

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 47.8%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 43.6%

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

   if -225 < z < 1.0999999999999999e-165

   1. Initial program 88.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 84.6

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

   5. Applied rewrites 84.6%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 84.9

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

   8. Applied rewrites 84.9%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 52.6%

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

   if 1.0999999999999999e-165 < z < 82000

   1. Initial program 88.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 56.7

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

   5. Applied rewrites 56.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 48.4%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 48.7%

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

   if 82000 < z

   1. Initial program 39.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 2.6%

       \[\leadsto 0 \cdot x \]

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 36.5%

       \[\leadsto 0 + \color{blue}{t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -225:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-165}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{elif}\;z \leq 82000:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -11500:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 2.05 \cdot 10^{+65}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -11500.0)
   (fma (/ x z) (- y a) t)
   (if (<= z 2.05e+65) (fma (- t x) (/ (- y z) a) x) (fma (/ (- x t) z) y t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -11500.0) {
		tmp = fma((x / z), (y - a), t);
	} else if (z <= 2.05e+65) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -11500.0)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	elseif (z <= 2.05e+65)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -11500.0], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 2.05e+65], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -11500

   1. Initial program 44.6%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.6%

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

   if -11500 < z < 2.0500000000000001e65

   1. Initial program 88.4%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 78.7

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

   5. Applied rewrites 78.7%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 82.0

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

   8. Applied rewrites 82.0%

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

   if 2.0500000000000001e65 < z

   1. Initial program 23.6%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 74.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.0%

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

Alternative 9: 62.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{+45} \lor \neg \left(a \leq 4.7 \cdot 10^{+61}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.7e+45) (not (<= a 4.7e+61)))
   (fma (/ (- x t) a) z x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.7e+45) || !(a <= 4.7e+61)) {
		tmp = fma(((x - t) / a), z, x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.7e+45) || !(a <= 4.7e+61))
		tmp = fma(Float64(Float64(x - t) / a), z, x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.7e+45], N[Not[LessEqual[a, 4.7e+61]], $MachinePrecision]], N[(N[(N[(x - t), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.69999999999999977e45 or 4.6999999999999998e61 < a

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 67.2

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

   5. Applied rewrites 67.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 60.4%

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

   if -3.69999999999999977e45 < a < 4.6999999999999998e61

   1. Initial program 63.5%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 74.9%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.0%

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

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{+45} \lor \neg \left(a \leq 4.7 \cdot 10^{+61}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7000:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 2.8 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7000.0)
   (fma (/ x z) (- y a) t)
   (if (<= z 2.8e+39) (fma (- t x) (/ y a) x) (fma (/ (- x t) z) y t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7000.0) {
		tmp = fma((x / z), (y - a), t);
	} else if (z <= 2.8e+39) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7000.0)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	elseif (z <= 2.8e+39)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7000.0], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 2.8e+39], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7e3

   1. Initial program 44.6%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.6%

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

   if -7e3 < z < 2.80000000000000001e39

   1. Initial program 88.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 80.4

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

   5. Applied rewrites 80.4%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 82.8

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

   8. Applied rewrites 82.8%

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

   9. Taylor expanded in y around inf

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

   11. Applied rewrites 79.0%

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

   if 2.80000000000000001e39 < z

   1. Initial program 27.9%

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

   3. Taylor expanded in z around inf

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

      1. fp-cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. associate--l- N/A

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

      5. fp-cancel-sign-sub N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. div-sub N/A

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

      9. *-lft-identity N/A

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

      10. metadata-eval N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. +-commutative N/A

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

   5. Applied rewrites 72.8%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 69.4%

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

Alternative 11: 40.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{+45} \lor \neg \left(a \leq 1.3 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.7e+45) (not (<= a 1.3e+41)))
   (fma x (/ z a) x)
   (fma a (/ t z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.7e+45) || !(a <= 1.3e+41)) {
		tmp = fma(x, (z / a), x);
	} else {
		tmp = fma(a, (t / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.7e+45) || !(a <= 1.3e+41))
		tmp = fma(x, Float64(z / a), x);
	else
		tmp = fma(a, Float64(t / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.7e+45], N[Not[LessEqual[a, 1.3e+41]], $MachinePrecision]], N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.69999999999999977e45 or 1.3e41 < a

   1. Initial program 67.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 49.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 48.7%

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

   if -3.69999999999999977e45 < a < 1.3e41

   1. Initial program 63.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 30.4

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

   5. Applied rewrites 30.4%

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

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 35.7%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 36.4%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{+45} \lor \neg \left(a \leq 1.3 \cdot 10^{+41}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 36.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -36 \lor \neg \left(z \leq 72000\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -36.0) (not (<= z 72000.0))) t (* t (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -36.0) || !(z <= 72000.0)) {
		tmp = t;
	} else {
		tmp = t * (y / a);
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-36.0d0)) .or. (.not. (z <= 72000.0d0))) then
        tmp = t
    else
        tmp = t * (y / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -36.0) || !(z <= 72000.0)) {
		tmp = t;
	} else {
		tmp = t * (y / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -36.0) or not (z <= 72000.0):
		tmp = t
	else:
		tmp = t * (y / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -36.0) || !(z <= 72000.0))
		tmp = t;
	else
		tmp = Float64(t * Float64(y / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -36.0) || ~((z <= 72000.0)))
		tmp = t;
	else
		tmp = t * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -36.0], N[Not[LessEqual[z, 72000.0]], $MachinePrecision]], t, N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -36 or 72000 < z

   1. Initial program 42.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 11.7%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 3.0%

       \[\leadsto 0 \cdot x \]

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 40.3%

       \[\leadsto 0 + \color{blue}{t} \]

   if -36 < z < 72000

   1. Initial program 88.6%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 82.5

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

   5. Applied rewrites 82.5%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 84.4

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

   8. Applied rewrites 84.4%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 32.7%

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

   12. Taylor expanded in y around inf

       \[\leadsto \frac{t \cdot y}{a} \]
   13. Step-by-step derivation

   14. Applied rewrites 34.6%

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

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -36 \lor \neg \left(z \leq 72000\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 34.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -43 \lor \neg \left(z \leq 1.7 \cdot 10^{-5}\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -43.0) (not (<= z 1.7e-5))) t (/ (* t y) a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -43.0) || !(z <= 1.7e-5)) {
		tmp = t;
	} else {
		tmp = (t * y) / a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-43.0d0)) .or. (.not. (z <= 1.7d-5))) then
        tmp = t
    else
        tmp = (t * y) / a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -43.0) || !(z <= 1.7e-5)) {
		tmp = t;
	} else {
		tmp = (t * y) / a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -43.0) or not (z <= 1.7e-5):
		tmp = t
	else:
		tmp = (t * y) / a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -43.0) || !(z <= 1.7e-5))
		tmp = t;
	else
		tmp = Float64(Float64(t * y) / a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -43.0) || ~((z <= 1.7e-5)))
		tmp = t;
	else
		tmp = (t * y) / a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -43.0], N[Not[LessEqual[z, 1.7e-5]], $MachinePrecision]], t, N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -43 or 1.7e-5 < z

   1. Initial program 42.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 48.8

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

   5. Applied rewrites 48.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 12.9%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 3.0%

       \[\leadsto 0 \cdot x \]

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 39.2%

       \[\leadsto 0 + \color{blue}{t} \]

   if -43 < z < 1.7e-5

   1. Initial program 89.8%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 29.5%

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

4. Final simplification 34.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -43 \lor \neg \left(z \leq 1.7 \cdot 10^{-5}\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 72000:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -0.74) (fma a (/ t z) t) (if (<= z 72000.0) (* t (/ y a)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -0.74) {
		tmp = fma(a, (t / z), t);
	} else if (z <= 72000.0) {
		tmp = t * (y / a);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -0.74)
		tmp = fma(a, Float64(t / z), t);
	elseif (z <= 72000.0)
		tmp = Float64(t * Float64(y / a));
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -0.74], N[(a * N[(t / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 72000.0], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], t]]

Derivation

1. Split input into 3 regimes

2. if z < -0.73999999999999999

   1. Initial program 43.5%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 47.8

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

   5. Applied rewrites 47.8%

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

   6. Taylor expanded in x around 0

      \[\leadsto -1 \cdot \color{blue}{\frac{t \cdot z}{a - z}} \]
   7. Step-by-step derivation

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 42.5%

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

   if -0.73999999999999999 < z < 72000

   1. Initial program 89.9%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 83.8

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

   5. Applied rewrites 83.8%

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

   6. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 84.1

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

   8. Applied rewrites 84.1%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 33.2%

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

   12. Taylor expanded in y around inf

       \[\leadsto \frac{t \cdot y}{a} \]
   13. Step-by-step derivation

   14. Applied rewrites 35.1%

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

   if 72000 < z

   1. Initial program 39.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. distribute-lft-neg-in N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 49.5

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

   5. Applied rewrites 49.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 12.5%

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

   9. Taylor expanded in z around inf

      \[\leadsto x + -1 \cdot \color{blue}{x} \]
   10. Step-by-step derivation

   11. Applied rewrites 2.6%

       \[\leadsto 0 \cdot x \]

   12. Taylor expanded in z around inf

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

   14. Applied rewrites 36.5%

       \[\leadsto 0 + \color{blue}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 37.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.74:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{t}{z}, t\right)\\ \mathbf{elif}\;z \leq 72000:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 25.0% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 45.0

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

5. Applied rewrites 45.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 23.4%

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

9. Taylor expanded in z around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
10. Step-by-step derivation

11. Applied rewrites 2.9%

    \[\leadsto 0 \cdot x \]

12. Taylor expanded in z around inf

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

14. Applied rewrites 23.4%

    \[\leadsto 0 + \color{blue}{t} \]

15. Final simplification 23.4%

    \[\leadsto t \]
16. Add Preprocessing

Alternative 16: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 64.8%

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

3. Taylor expanded in y around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

   5. distribute-lft-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower--.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lower--.f64 45.0

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

5. Applied rewrites 45.0%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 23.4%

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

9. Taylor expanded in z around inf

   \[\leadsto x + -1 \cdot \color{blue}{x} \]
10. Step-by-step derivation

11. Applied rewrites 2.9%

    \[\leadsto 0 \cdot x \]

12. Taylor expanded in x around 0

    \[\leadsto 0 \]
13. Step-by-step derivation

14. Applied rewrites 2.9%

    \[\leadsto 0 \]
15. Add Preprocessing

Developer Target 1: 83.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - x)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2025015 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))