Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 97.0% → 96.8%

Time: 3.5s

Alternatives: 14

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 10:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 10.0) (/ (* (- x y) t_m) (- z y)) (* (- x y) (/ t_m (- z y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 10.0) {
		tmp = ((x - y) * t_m) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 10.0d0) then
        tmp = ((x - y) * t_m) / (z - y)
    else
        tmp = (x - y) * (t_m / (z - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 10.0) {
		tmp = ((x - y) * t_m) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 10.0:
		tmp = ((x - y) * t_m) / (z - y)
	else:
		tmp = (x - y) * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 10.0)
		tmp = Float64(Float64(Float64(x - y) * t_m) / Float64(z - y));
	else
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 10.0)
		tmp = ((x - y) * t_m) / (z - y);
	else
		tmp = (x - y) * (t_m / (z - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 10.0], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 10

   1. Initial program 95.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 91.5

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

   4. Applied rewrites 91.5%

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

   if 10 < t

   1. Initial program 96.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 66.2

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

   4. Applied rewrites 66.2%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.8

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

   6. Applied rewrites 99.8%

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

Alternative 2: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x \cdot t\_m}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{y - x}{y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (* x t_m) (- z y))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -2e+18)
      t_2
      (if (<= t_3 1e-5)
        (* (/ (- x y) z) t_m)
        (if (<= t_3 2.0) (* (/ (- y x) y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x * t_m) / (z - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -2e+18) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (x * t_m) / (z - y)
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-2d+18)) then
        tmp = t_2
    else if (t_3 <= 1d-5) then
        tmp = ((x - y) / z) * t_m
    else if (t_3 <= 2.0d0) then
        tmp = ((y - x) / y) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x * t_m) / (z - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -2e+18) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x * t_m) / (z - y)
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -2e+18:
		tmp = t_2
	elif t_3 <= 1e-5:
		tmp = ((x - y) / z) * t_m
	elif t_3 <= 2.0:
		tmp = ((y - x) / y) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x * t_m) / Float64(z - y))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -2e+18)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(Float64(y - x) / y) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x * t_m) / (z - y);
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -2e+18)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = ((x - y) / z) * t_m;
	elseif (t_3 <= 2.0)
		tmp = ((y - x) / y) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x * t$95$m), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -2e+18], t$95$2, If[LessEqual[t$95$3, 1e-5], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e18 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 92.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 97.4

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

   4. Applied rewrites 97.4%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 97.2%

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

   if -2e18 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 94.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.5%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 98.7

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

   11. Applied rewrites 98.7%

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

Alternative 3: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z - y} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{y - x}{y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ x (- z y)) t_m)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.2)
      t_2
      (if (<= t_3 1e-5)
        (* (/ (- x y) z) t_m)
        (if (<= t_3 2.0) (* (/ (- y x) y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x / (z - y)) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.2) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (x / (z - y)) * t_m
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-0.2d0)) then
        tmp = t_2
    else if (t_3 <= 1d-5) then
        tmp = ((x - y) / z) * t_m
    else if (t_3 <= 2.0d0) then
        tmp = ((y - x) / y) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x / (z - y)) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.2) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_3 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x / (z - y)) * t_m
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -0.2:
		tmp = t_2
	elif t_3 <= 1e-5:
		tmp = ((x - y) / z) * t_m
	elif t_3 <= 2.0:
		tmp = ((y - x) / y) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x / Float64(z - y)) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.2)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(Float64(y - x) / y) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x / (z - y)) * t_m;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -0.2)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = ((x - y) / z) * t_m;
	elseif (t_3 <= 2.0)
		tmp = ((y - x) / y) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.2], t$95$2, If[LessEqual[t$95$3, 1e-5], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -0.20000000000000001 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.7%

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

   if -0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 94.4%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 92.2%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 98.7

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

   11. Applied rewrites 98.7%

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

Alternative 4: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z - y} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{-5}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\frac{y - x}{y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ x (- z y)) t_m)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.2)
      t_2
      (if (<= t_3 1e-5)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0) (* (/ (- y x) y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x / (z - y)) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.2) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (x / (z - y)) * t_m
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-0.2d0)) then
        tmp = t_2
    else if (t_3 <= 1d-5) then
        tmp = (x - y) * (t_m / z)
    else if (t_3 <= 2.0d0) then
        tmp = ((y - x) / y) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x / (z - y)) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.2) {
		tmp = t_2;
	} else if (t_3 <= 1e-5) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x / (z - y)) * t_m
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -0.2:
		tmp = t_2
	elif t_3 <= 1e-5:
		tmp = (x - y) * (t_m / z)
	elif t_3 <= 2.0:
		tmp = ((y - x) / y) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x / Float64(z - y)) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.2)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = Float64(Float64(Float64(y - x) / y) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x / (z - y)) * t_m;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -0.2)
		tmp = t_2;
	elseif (t_3 <= 1e-5)
		tmp = (x - y) * (t_m / z);
	elseif (t_3 <= 2.0)
		tmp = ((y - x) / y) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.2], t$95$2, If[LessEqual[t$95$3, 1e-5], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -0.20000000000000001 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 91.7%

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

   if -0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 94.4%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 86.7

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

   5. Applied rewrites 86.7%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 89.4

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

   7. Applied rewrites 89.4%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 98.7

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

   11. Applied rewrites 98.7%

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

Alternative 5: 79.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -4 \cdot 10^{-31}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{y - x}{y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -4e-31)
      (/ (* (- x y) t_m) z)
      (if (<= t_2 1e-5)
        (* (- x y) (/ t_m z))
        (if (<= t_2 2.0) (* (/ (- y x) y) t_m) (/ (* t_m x) z)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -4e-31) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_2 <= 1e-5) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-4d-31)) then
        tmp = ((x - y) * t_m) / z
    else if (t_2 <= 1d-5) then
        tmp = (x - y) * (t_m / z)
    else if (t_2 <= 2.0d0) then
        tmp = ((y - x) / y) * t_m
    else
        tmp = (t_m * x) / z
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -4e-31) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_2 <= 1e-5) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -4e-31:
		tmp = ((x - y) * t_m) / z
	elif t_2 <= 1e-5:
		tmp = (x - y) * (t_m / z)
	elif t_2 <= 2.0:
		tmp = ((y - x) / y) * t_m
	else:
		tmp = (t_m * x) / z
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -4e-31)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_2 <= 1e-5)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(Float64(y - x) / y) * t_m);
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -4e-31)
		tmp = ((x - y) * t_m) / z;
	elseif (t_2 <= 1e-5)
		tmp = (x - y) * (t_m / z);
	elseif (t_2 <= 2.0)
		tmp = ((y - x) / y) * t_m;
	else
		tmp = (t_m * x) / z;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -4e-31], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 1e-5], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -4e-31

   1. Initial program 95.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 62.4

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

   5. Applied rewrites 62.4%

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

   if -4e-31 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 93.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 87.5

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

   5. Applied rewrites 87.5%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 93.6

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

   7. Applied rewrites 93.6%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 98.7

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

   11. Applied rewrites 98.7%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

Alternative 6: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x \cdot t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{x}{z - y}, t\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -2e+18)
      (/ (* x t_m) (- z y))
      (if (<= t_2 1e-5)
        (* (- x y) (/ t_m (- z y)))
        (fma t_m (/ x (- z y)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -2e+18) {
		tmp = (x * t_m) / (z - y);
	} else if (t_2 <= 1e-5) {
		tmp = (x - y) * (t_m / (z - y));
	} else {
		tmp = fma(t_m, (x / (z - y)), t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -2e+18)
		tmp = Float64(Float64(x * t_m) / Float64(z - y));
	elseif (t_2 <= 1e-5)
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	else
		tmp = fma(t_m, Float64(x / Float64(z - y)), t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -2e+18], N[(N[(x * t$95$m), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-5], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] + t$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e18

   1. Initial program 94.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.7%

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

   if -2e18 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 94.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 86.8

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

   4. Applied rewrites 86.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 92.0

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

   6. Applied rewrites 92.0%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 95.6%

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

Alternative 7: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+18}:\\ \;\;\;\;\frac{x \cdot t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{x}{z - y}, t\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -2e+18)
      (/ (* x t_m) (- z y))
      (if (<= t_2 1e-5) (* (/ (- x y) z) t_m) (fma t_m (/ x (- z y)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -2e+18) {
		tmp = (x * t_m) / (z - y);
	} else if (t_2 <= 1e-5) {
		tmp = ((x - y) / z) * t_m;
	} else {
		tmp = fma(t_m, (x / (z - y)), t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -2e+18)
		tmp = Float64(Float64(x * t_m) / Float64(z - y));
	elseif (t_2 <= 1e-5)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	else
		tmp = fma(t_m, Float64(x / Float64(z - y)), t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -2e+18], N[(N[(x * t$95$m), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e-5], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] + t$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e18

   1. Initial program 94.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 99.7

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

   4. Applied rewrites 99.7%

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

   5. Taylor expanded in x around inf

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

   7. Applied rewrites 99.7%

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

   if -2e18 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 94.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 90.5%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lift--.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 N/A

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

      8. associate-/l* N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 N/A

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

      11. lift--.f64 97.1

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

   5. Applied rewrites 97.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 95.6%

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

Alternative 8: 79.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-5}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{y - x}{y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 1e-5)
      (* (- x y) (/ t_m z))
      (if (<= t_2 2.0) (* (/ (- y x) y) t_m) (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-5) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 1d-5) then
        tmp = (x - y) * (t_m / z)
    else if (t_2 <= 2.0d0) then
        tmp = ((y - x) / y) * t_m
    else
        tmp = (t_m * x) / z
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-5) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 1e-5:
		tmp = (x - y) * (t_m / z)
	elif t_2 <= 2.0:
		tmp = ((y - x) / y) * t_m
	else:
		tmp = (t_m * x) / z
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 1e-5)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(Float64(y - x) / y) * t_m);
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 1e-5)
		tmp = (x - y) * (t_m / z);
	elseif (t_2 <= 2.0)
		tmp = ((y - x) / y) * t_m;
	else
		tmp = (t_m * x) / z;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 1e-5], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000008e-5

   1. Initial program 94.5%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 77.9

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

   5. Applied rewrites 77.9%

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

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 N/A

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

      7. lower-/.f64 77.6

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

   7. Applied rewrites 77.6%

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

   if 1.00000000000000008e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 68.7

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

   5. Applied rewrites 68.7%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 98.7

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

   8. Applied rewrites 98.7%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 98.7

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

   11. Applied rewrites 98.7%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

Alternative 9: 70.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-9}:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{y - x}{y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 1e-9)
      (* (/ x z) t_m)
      (if (<= t_2 2.0) (* (/ (- y x) y) t_m) (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-9) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 1d-9) then
        tmp = (x / z) * t_m
    else if (t_2 <= 2.0d0) then
        tmp = ((y - x) / y) * t_m
    else
        tmp = (t_m * x) / z
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-9) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = ((y - x) / y) * t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 1e-9:
		tmp = (x / z) * t_m
	elif t_2 <= 2.0:
		tmp = ((y - x) / y) * t_m
	else:
		tmp = (t_m * x) / z
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 1e-9)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(Float64(y - x) / y) * t_m);
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 1e-9)
		tmp = (x / z) * t_m;
	elseif (t_2 <= 2.0)
		tmp = ((y - x) / y) * t_m;
	else
		tmp = (t_m * x) / z;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 1e-9], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000006e-9

   1. Initial program 94.5%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.00000000000000006e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lift--.f64 68.1

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

   5. Applied rewrites 68.1%

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

   6. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 97.8

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

   8. Applied rewrites 97.8%

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

   9. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 97.8

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

   11. Applied rewrites 97.8%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

Alternative 10: 68.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-9} \lor \neg \left(t\_2 \leq 2\right):\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (* t_s (if (or (<= t_2 1e-9) (not (<= t_2 2.0))) (/ (* t_m x) z) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0)) {
		tmp = (t_m * x) / z;
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if ((t_2 <= 1d-9) .or. (.not. (t_2 <= 2.0d0))) then
        tmp = (t_m * x) / z
    else
        tmp = t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0)) {
		tmp = (t_m * x) / z;
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if (t_2 <= 1e-9) or not (t_2 <= 2.0):
		tmp = (t_m * x) / z
	else:
		tmp = t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0))
		tmp = Float64(Float64(t_m * x) / z);
	else
		tmp = t_m;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_2 <= 1e-9) || ~((t_2 <= 2.0)))
		tmp = (t_m * x) / z;
	else
		tmp = t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 1e-9], N[Not[LessEqual[t$95$2, 2.0]], $MachinePrecision]], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision], t$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000006e-9 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 58.6

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

   5. Applied rewrites 58.6%

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

   if 1.00000000000000006e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.0%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-9} \lor \neg \left(\frac{x - y}{z - y} \leq 2\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 68.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-9} \lor \neg \left(t\_2 \leq 2\right):\\ \;\;\;\;x \cdot \frac{t\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (* t_s (if (or (<= t_2 1e-9) (not (<= t_2 2.0))) (* x (/ t_m z)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0)) {
		tmp = x * (t_m / z);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if ((t_2 <= 1d-9) .or. (.not. (t_2 <= 2.0d0))) then
        tmp = x * (t_m / z)
    else
        tmp = t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0)) {
		tmp = x * (t_m / z);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if (t_2 <= 1e-9) or not (t_2 <= 2.0):
		tmp = x * (t_m / z)
	else:
		tmp = t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_2 <= 1e-9) || !(t_2 <= 2.0))
		tmp = Float64(x * Float64(t_m / z));
	else
		tmp = t_m;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_2 <= 1e-9) || ~((t_2 <= 2.0)))
		tmp = x * (t_m / z);
	else
		tmp = t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 1e-9], N[Not[LessEqual[t$95$2, 2.0]], $MachinePrecision]], N[(x * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000006e-9 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 73.0

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

   5. Applied rewrites 73.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 58.6%

      \[\leadsto \frac{x \cdot t}{z} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 55.8

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

   10. Applied rewrites 55.8%

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

   if 1.00000000000000006e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.0%

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

4. Final simplification 69.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-9} \lor \neg \left(\frac{x - y}{z - y} \leq 2\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-9}:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 1e-9) (* (/ x z) t_m) (if (<= t_2 2.0) t_m (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-9) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 1d-9) then
        tmp = (x / z) * t_m
    else if (t_2 <= 2.0d0) then
        tmp = t_m
    else
        tmp = (t_m * x) / z
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-9) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 1e-9:
		tmp = (x / z) * t_m
	elif t_2 <= 2.0:
		tmp = t_m
	else:
		tmp = (t_m * x) / z
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 1e-9)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 1e-9)
		tmp = (x / z) * t_m;
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = (t_m * x) / z;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 1e-9], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], t$95$m, N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000006e-9

   1. Initial program 94.5%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

   if 1.00000000000000006e-9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 95.0%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.6%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 59.6

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

   5. Applied rewrites 59.6%

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

Alternative 13: 97.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \left(\frac{x - y}{z - y} \cdot t\_m\right) \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (* t_s (* (/ (- x y) (- z y)) t_m)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (((x - y) / (z - y)) * t_m);
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * (((x - y) / (z - y)) * t_m)
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (((x - y) / (z - y)) * t_m);
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	return t_s * (((x - y) / (z - y)) * t_m)

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	return Float64(t_s * Float64(Float64(Float64(x - y) / Float64(z - y)) * t_m))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * (((x - y) / (z - y)) * t_m);
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * N[(N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.9%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 14: 35.0% accurate, 23.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot t\_m \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m) :precision binary64 (* t_s t_m))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * t_m;
}

Fortran

t\_m =     private
t\_s =     private
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(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * t_m
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * t_m;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	return t_s * t_m

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	return Float64(t_s * t_m)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * t_m;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * t$95$m), $MachinePrecision]

Derivation

1. Initial program 95.9%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

5. Applied rewrites 35.6%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Developer Target 1: 97.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{t}{\frac{z - y}{x - y}} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025051 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ t (/ (- z y) (- x y))))

  (* (/ (- x y) (- z y)) t))