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

Percentage Accurate: 96.7% → 96.7%

Time: 5.0s

Alternatives: 13

Speedup: 0.7×

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: 96.7% 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: 96.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 3.5 \cdot 10^{-43}:\\ \;\;\;\;\frac{-1}{z - y} \cdot \left(\left(y - x\right) \cdot t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot \left(x - y\right)\\ \end{array} \end{array} \]

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 3.5e-43)
    (* (/ -1.0 (- z y)) (* (- y x) t_m))
    (* (/ t_m (- z y)) (- x 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 <= 3.5e-43) {
		tmp = (-1.0 / (z - y)) * ((y - x) * t_m);
	} else {
		tmp = (t_m / (z - y)) * (x - 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 <= 3.5d-43) then
        tmp = ((-1.0d0) / (z - y)) * ((y - x) * t_m)
    else
        tmp = (t_m / (z - y)) * (x - 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 <= 3.5e-43) {
		tmp = (-1.0 / (z - y)) * ((y - x) * t_m);
	} else {
		tmp = (t_m / (z - y)) * (x - 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 <= 3.5e-43:
		tmp = (-1.0 / (z - y)) * ((y - x) * t_m)
	else:
		tmp = (t_m / (z - y)) * (x - 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 <= 3.5e-43)
		tmp = Float64(Float64(-1.0 / Float64(z - y)) * Float64(Float64(y - x) * t_m));
	else
		tmp = Float64(Float64(t_m / Float64(z - y)) * Float64(x - 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 <= 3.5e-43)
		tmp = (-1.0 / (z - y)) * ((y - x) * t_m);
	else
		tmp = (t_m / (z - y)) * (x - 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, 3.5e-43], N[(N[(-1.0 / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] * t$95$m), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 3.49999999999999997e-43

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. mult-flip N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg-rev N/A

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

      11. lower-/.f64 N/A

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

      12. *-commutative N/A

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

      13. lower-*.f64 N/A

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

      14. lift--.f64 N/A

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

      15. sub-negate-rev N/A

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

      16. lower--.f64 83.8

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

   3. Applied rewrites 83.8%

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

   if 3.49999999999999997e-43 < t

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mult-flip N/A

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

      9. lower-/.f64 84.7

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

   3. Applied rewrites 84.7%

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

Alternative 2: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-223}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

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-223)
      (* (/ t_m (- z y)) (- x y))
      (if (<= t_2 2e-6)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0)
          (fma t_m (/ (- z x) y) t_m)
          (/ (* t_m x) (- 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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-223) {
		tmp = (t_m / (z - y)) * (x - y);
	} else if (t_2 <= 2e-6) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, ((z - x) / y), t_m);
	} else {
		tmp = (t_m * x) / (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)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 1e-223)
		tmp = Float64(Float64(t_m / Float64(z - y)) * Float64(x - y));
	elseif (t_2 <= 2e-6)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(Float64(z - x) / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	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, 1e-223], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2e-6], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999997e-224

   1. Initial program 96.7%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. mult-flip N/A

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

      4. associate-*l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. mult-flip N/A

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

      9. lower-/.f64 84.7

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

   3. Applied rewrites 84.7%

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

   if 9.9999999999999997e-224 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.5%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.5

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

   4. Applied rewrites 47.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out-- N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 51.5

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

   6. Applied rewrites 51.5%

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

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

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

Alternative 3: 94.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

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-17)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 2e-6)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0)
          (fma t_m (/ (- z x) y) t_m)
          (/ (* t_m x) (- 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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -1e-17) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-6) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, ((z - x) / y), t_m);
	} else {
		tmp = (t_m * x) / (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)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -1e-17)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 2e-6)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(Float64(z - x) / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	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, -1e-17], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e-6], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000007e-17

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if -1.00000000000000007e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.5%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.5

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

   4. Applied rewrites 47.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lift-/.f64 N/A

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

      6. distribute-neg-frac N/A

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

      7. lift--.f64 N/A

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

      8. sub-negate-rev N/A

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

      9. lift-*.f64 N/A

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

      10. lift-*.f64 N/A

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

      11. distribute-lft-out-- N/A

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

      12. associate-/l* N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 51.5

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

   6. Applied rewrites 51.5%

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

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

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

Alternative 4: 93.3% 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 96.7%

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

Alternative 5: 92.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 1.001:\\ \;\;\;\;t\_m + \frac{t\_m \cdot z}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

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-17)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 2e-6)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 1.001) (+ t_m (/ (* t_m z) y)) (/ (* t_m x) (- 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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -1e-17) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-6) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 1.001) {
		tmp = t_m + ((t_m * z) / y);
	} else {
		tmp = (t_m * x) / (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) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-1d-17)) then
        tmp = (t_m / (z - y)) * x
    else if (t_2 <= 2d-6) then
        tmp = ((x - y) / z) * t_m
    else if (t_2 <= 1.001d0) then
        tmp = t_m + ((t_m * z) / y)
    else
        tmp = (t_m * x) / (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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -1e-17) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-6) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 1.001) {
		tmp = t_m + ((t_m * z) / y);
	} else {
		tmp = (t_m * x) / (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):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -1e-17:
		tmp = (t_m / (z - y)) * x
	elif t_2 <= 2e-6:
		tmp = ((x - y) / z) * t_m
	elif t_2 <= 1.001:
		tmp = t_m + ((t_m * z) / y)
	else:
		tmp = (t_m * x) / (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)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -1e-17)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 2e-6)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 1.001)
		tmp = Float64(t_m + Float64(Float64(t_m * z) / y));
	else
		tmp = Float64(Float64(t_m * x) / 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)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -1e-17)
		tmp = (t_m / (z - y)) * x;
	elseif (t_2 <= 2e-6)
		tmp = ((x - y) / z) * t_m;
	elseif (t_2 <= 1.001)
		tmp = t_m + ((t_m * z) / y);
	else
		tmp = (t_m * x) / (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_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -1e-17], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e-6], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 1.001], N[(t$95$m + N[(N[(t$95$m * z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000007e-17

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if -1.00000000000000007e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.5%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.0009999999999999

   1. Initial program 96.7%

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

   2. Taylor expanded in y around -inf

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

      1. lower-+.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower-*.f64 47.5

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

   4. Applied rewrites 47.5%

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

   5. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 33.8

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

   7. Applied rewrites 33.8%

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

   if 1.0009999999999999 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

Alternative 6: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-17}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

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-17)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 2e-6)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0) t_m (/ (* t_m x) (- 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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -1e-17) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-6) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / (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) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-1d-17)) then
        tmp = (t_m / (z - y)) * x
    else if (t_2 <= 2d-6) then
        tmp = ((x - y) / z) * t_m
    else if (t_2 <= 2.0d0) then
        tmp = t_m
    else
        tmp = (t_m * x) / (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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -1e-17) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-6) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / (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):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -1e-17:
		tmp = (t_m / (z - y)) * x
	elif t_2 <= 2e-6:
		tmp = ((x - y) / z) * t_m
	elif t_2 <= 2.0:
		tmp = t_m
	else:
		tmp = (t_m * x) / (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)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -1e-17)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 2e-6)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = Float64(Float64(t_m * x) / 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)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -1e-17)
		tmp = (t_m / (z - y)) * x;
	elseif (t_2 <= 2e-6)
		tmp = ((x - y) / z) * t_m;
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = (t_m * x) / (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_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -1e-17], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e-6], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], t$95$m, N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1.00000000000000007e-17

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if -1.00000000000000007e-17 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.5%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 34.9%

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

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

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

Alternative 7: 91.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1 \cdot 10^{-7}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

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-7)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 2e-6)
        (/ (* t_m (- x y)) z)
        (if (<= t_2 2.0) t_m (/ (* t_m x) (- 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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -1e-7) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-6) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / (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) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-1d-7)) then
        tmp = (t_m / (z - y)) * x
    else if (t_2 <= 2d-6) then
        tmp = (t_m * (x - y)) / z
    else if (t_2 <= 2.0d0) then
        tmp = t_m
    else
        tmp = (t_m * x) / (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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -1e-7) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-6) {
		tmp = (t_m * (x - y)) / z;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / (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):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -1e-7:
		tmp = (t_m / (z - y)) * x
	elif t_2 <= 2e-6:
		tmp = (t_m * (x - y)) / z
	elif t_2 <= 2.0:
		tmp = t_m
	else:
		tmp = (t_m * x) / (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)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -1e-7)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 2e-6)
		tmp = Float64(Float64(t_m * Float64(x - y)) / z);
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = Float64(Float64(t_m * x) / 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)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -1e-7)
		tmp = (t_m / (z - y)) * x;
	elseif (t_2 <= 2e-6)
		tmp = (t_m * (x - y)) / z;
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = (t_m * x) / (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_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -1e-7], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e-6], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 2.0], t$95$m, N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -9.9999999999999995e-8

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if -9.9999999999999995e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6

   1. Initial program 96.7%

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

   2. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.6

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

   4. Applied rewrites 47.6%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 34.9%

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

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

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

Alternative 8: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

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-6)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 2.0) t_m (/ (* t_m x) (- 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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 2e-6) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / (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) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 2d-6) then
        tmp = (t_m / (z - y)) * x
    else if (t_2 <= 2.0d0) then
        tmp = t_m
    else
        tmp = (t_m * x) / (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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 2e-6) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / (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):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 2e-6:
		tmp = (t_m / (z - y)) * x
	elif t_2 <= 2.0:
		tmp = t_m
	else:
		tmp = (t_m * x) / (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)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 2e-6)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = Float64(Float64(t_m * x) / 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)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 2e-6)
		tmp = (t_m / (z - y)) * x;
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = (t_m * x) / (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_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 2e-6], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2.0], t$95$m, N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 34.9%

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

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

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

Alternative 9: 81.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 2e-6) t_2 (if (<= t_3 2.0) 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 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-6) then
        tmp = t_2
    else if (t_3 <= 2.0d0) then
        tmp = 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 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-6:
		tmp = t_2
	elif t_3 <= 2.0:
		tmp = 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(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $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-6], t$95$2, If[LessEqual[t$95$3, 2.0], t$95$m, t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.7%

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

   2. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 50.3

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

   4. Applied rewrites 50.3%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 50.5

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

   6. Applied rewrites 50.5%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 34.9%

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

Alternative 10: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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) t_m)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 2e-6) t_2 (if (<= t_3 2.0) 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) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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) * t_m
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-6) then
        tmp = t_2
    else if (t_3 <= 2.0d0) then
        tmp = 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) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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) * t_m
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-6:
		tmp = t_2
	elif t_3 <= 2.0:
		tmp = 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 / z) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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) * t_m;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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 / z), $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, 2e-6], t$95$2, If[LessEqual[t$95$3, 2.0], t$95$m, t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 39.7

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

   4. Applied rewrites 39.7%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 34.9%

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

Alternative 11: 68.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m \cdot x}{z}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (/ (* t_m x) z)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 2e-6) t_2 (if (<= t_3 2.0) 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 = (t_m * x) / z;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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 = (t_m * x) / z
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-6) then
        tmp = t_2
    else if (t_3 <= 2.0d0) then
        tmp = 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 = (t_m * x) / z;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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 = (t_m * x) / z
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-6:
		tmp = t_2
	elif t_3 <= 2.0:
		tmp = 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(t_m * x) / z)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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 = (t_m * x) / z;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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[(t$95$m * x), $MachinePrecision] / z), $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-6], t$95$2, If[LessEqual[t$95$3, 2.0], t$95$m, t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 34.9%

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

Alternative 12: 68.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (* (/ t_m z) x)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 2e-6) t_2 (if (<= t_3 2.0) 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 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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 = (t_m / z) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-6) then
        tmp = t_2
    else if (t_3 <= 2.0d0) then
        tmp = 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 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-6) {
		tmp = t_2;
	} else if (t_3 <= 2.0) {
		tmp = 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 = (t_m / z) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-6:
		tmp = t_2
	elif t_3 <= 2.0:
		tmp = 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(t_m / z) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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 = (t_m / z) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-6)
		tmp = t_2;
	elseif (t_3 <= 2.0)
		tmp = 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[(t$95$m / z), $MachinePrecision] * x), $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-6], t$95$2, If[LessEqual[t$95$3, 2.0], t$95$m, t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999991e-6 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.7%

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

   2. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 37.7

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

   4. Applied rewrites 37.7%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

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

      9. lower-/.f64 37.8

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

   6. Applied rewrites 37.8%

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

   if 1.99999999999999991e-6 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 96.7%

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

   2. Taylor expanded in y around inf

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

   4. Applied rewrites 34.9%

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

Alternative 13: 34.9% accurate, 12.6× 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 96.7%

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

2. Taylor expanded in y around inf

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

4. Applied rewrites 34.9%

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

Reproduce

herbie shell --seed 2025143 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64
  (* (/ (- x y) (- z y)) t))