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

Percentage Accurate: 97.0% → 97.1%

Time: 3.6s

Alternatives: 18

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 0.058:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 0.058) (/ (* (- x y) t_m) (- z y)) (* (- x y) (/ t_m (- z y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 0.058) {
		tmp = ((x - y) * t_m) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 0.058d0) then
        tmp = ((x - y) * t_m) / (z - y)
    else
        tmp = (x - y) * (t_m / (z - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 0.058) {
		tmp = ((x - y) * t_m) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 0.058:
		tmp = ((x - y) * t_m) / (z - y)
	else:
		tmp = (x - y) * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 0.058)
		tmp = Float64(Float64(Float64(x - y) * t_m) / Float64(z - y));
	else
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 0.058)
		tmp = ((x - y) * t_m) / (z - y);
	else
		tmp = (x - y) * (t_m / (z - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 0.058], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 0.0580000000000000029

   1. Initial program 96.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 88.9

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

   4. Applied rewrites 88.9%

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

   if 0.0580000000000000029 < t

   1. Initial program 96.8%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 76.6

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

   4. Applied rewrites 76.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 96.8

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

   6. Applied rewrites 96.8%

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

Alternative 2: 70.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 10^{-84}:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{-y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(-t\_m\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 1e-84)
      (* (/ x z) t_m)
      (if (<= t_2 0.2)
        (* (/ (- y) z) t_m)
        (if (<= t_2 20000000.0)
          (fma t_m (/ z y) t_m)
          (if (<= t_2 2e+77) (* (- t_m) (/ x y)) (/ (* t_m x) z))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 1e-84) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 0.2) {
		tmp = (-y / z) * t_m;
	} else if (t_2 <= 20000000.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_2 <= 2e+77) {
		tmp = -t_m * (x / y);
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 1e-84)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(-y) / z) * t_m);
	elseif (t_2 <= 20000000.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_2 <= 2e+77)
		tmp = Float64(Float64(-t_m) * Float64(x / y));
	else
		tmp = Float64(Float64(t_m * x) / z);
	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-84], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[((-y) / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 20000000.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+77], N[((-t$95$m) * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 5 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1e-84

   1. Initial program 93.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 68.5

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

   5. Applied rewrites 68.5%

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

   if 1e-84 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 94.3%

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

   6. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 64.6

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

   8. Applied rewrites 64.6%

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

   if 0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e7

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.2%

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

   if 2e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999997e77

   1. Initial program 99.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 70.4

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

   12. Applied rewrites 70.4%

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

   if 1.99999999999999997e77 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

      \[\leadsto \color{blue}{\frac{t \cdot x}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 10^{-84}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.2:\\ \;\;\;\;\frac{-y}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(-t\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.001:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{-y}{z - y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5e+19)
      (/ (* x t_m) (- z y))
      (if (<= t_2 0.001)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0) (* (/ (- y) (- z y)) t_m) (* (/ x (- z y)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+19) {
		tmp = (x * t_m) / (z - y);
	} else if (t_2 <= 0.001) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = (-y / (z - y)) * t_m;
	} else {
		tmp = (x / (z - y)) * t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-5d+19)) then
        tmp = (x * t_m) / (z - y)
    else if (t_2 <= 0.001d0) then
        tmp = ((x - y) / z) * t_m
    else if (t_2 <= 2.0d0) then
        tmp = (-y / (z - y)) * t_m
    else
        tmp = (x / (z - y)) * t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+19) {
		tmp = (x * t_m) / (z - y);
	} else if (t_2 <= 0.001) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = (-y / (z - y)) * t_m;
	} else {
		tmp = (x / (z - y)) * t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -5e+19:
		tmp = (x * t_m) / (z - y)
	elif t_2 <= 0.001:
		tmp = ((x - y) / z) * t_m
	elif t_2 <= 2.0:
		tmp = (-y / (z - y)) * t_m
	else:
		tmp = (x / (z - y)) * t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5e+19)
		tmp = Float64(Float64(x * t_m) / Float64(z - y));
	elseif (t_2 <= 0.001)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(Float64(-y) / Float64(z - y)) * t_m);
	else
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -5e+19)
		tmp = (x * t_m) / (z - y);
	elseif (t_2 <= 0.001)
		tmp = ((x - y) / z) * t_m;
	elseif (t_2 <= 2.0)
		tmp = (-y / (z - y)) * t_m;
	else
		tmp = (x / (z - y)) * t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e+19], N[(N[(x * t$95$m), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.001], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[((-y) / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e19

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -5e19 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1e-3

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 93.6%

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

   if 1e-3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.4

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

   5. Applied rewrites 99.4%

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

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

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

Alternative 4: 94.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+19}:\\ \;\;\;\;\frac{x \cdot t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5e+19)
      (/ (* x t_m) (- z y))
      (if (<= t_2 0.2)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (* (/ x (- z y)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+19) {
		tmp = (x * t_m) / (z - y);
	} else if (t_2 <= 0.2) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (x / (z - y)) * t_m;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5e+19)
		tmp = Float64(Float64(x * t_m) / Float64(z - y));
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e+19], N[(N[(x * t$95$m), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e19

   1. Initial program 90.2%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 90.2%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 96.7

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

   7. Applied rewrites 96.7%

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

   if -5e19 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 92.8%

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

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

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.4%

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

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

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

Alternative 5: 94.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -2000000000000:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -2000000000000.0)
      (* x (/ t_m (- z y)))
      (if (<= t_2 0.2)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (* (/ x (- z y)) t_m)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -2000000000000.0) {
		tmp = x * (t_m / (z - y));
	} else if (t_2 <= 0.2) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (x / (z - y)) * t_m;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -2000000000000.0)
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	elseif (t_2 <= 0.2)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -2000000000000.0], N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.2], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e12

   1. Initial program 90.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 90.3%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 93.6

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

   7. Applied rewrites 93.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 90.9

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

   9. Applied rewrites 90.9%

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

   if -2e12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001

   1. Initial program 95.7%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 93.7%

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

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

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.4%

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

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

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

Alternative 6: 93.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z - y} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -500:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ x (- z y)) t_m)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -500.0)
      t_2
      (if (<= t_3 0.2)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0) (fma t_m (/ z y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x / (z - y)) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -500.0) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x / Float64(z - y)) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -500.0)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = t_2;
	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 / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -500.0], t$95$2, If[LessEqual[t$95$3, 0.2], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -500 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.9%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.7%

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

   if -500 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001

   1. Initial program 95.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 89.6

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

   4. Applied rewrites 89.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 94.0

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

   6. Applied rewrites 94.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 91.9%

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

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

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.4%

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

Alternative 7: 91.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \frac{t\_m}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* x (/ t_m (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -2000000000000.0)
      t_2
      (if (<= t_3 0.2)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0) (fma t_m (/ z y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = x * (t_m / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(x * Float64(t_m / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = t_2;
	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[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -2000000000000.0], t$95$2, If[LessEqual[t$95$3, 0.2], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e12 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 92.9

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

   7. Applied rewrites 92.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 92.8

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

   9. Applied rewrites 92.8%

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

   if -2e12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001

   1. Initial program 95.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 89.8

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

   4. Applied rewrites 89.8%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 93.0

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

   6. Applied rewrites 93.0%

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

   7. Taylor expanded in y around 0

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

   9. Applied rewrites 91.1%

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

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

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.4%

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

Alternative 8: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \frac{t\_m}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -2000000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* x (/ t_m (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -2000000000000.0)
      t_2
      (if (<= t_3 0.2)
        (/ (* (- x y) t_m) z)
        (if (<= t_3 2.0) (fma t_m (/ z y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = x * (t_m / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -2000000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 0.2) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(x * Float64(t_m / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -2000000000000.0)
		tmp = t_2;
	elseif (t_3 <= 0.2)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = t_2;
	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[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -2000000000000.0], t$95$2, If[LessEqual[t$95$3, 0.2], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e12 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 93.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 92.9

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

   7. Applied rewrites 92.9%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lift-/.f64 N/A

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

      7. lift--.f64 92.8

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

   9. Applied rewrites 92.8%

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

   if -2e12 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001

   1. Initial program 95.7%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 89.5

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

   5. Applied rewrites 89.5%

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

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

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.4%

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

Alternative 9: 80.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{\left(x - y\right) \cdot t\_m}{z}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.2:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(-t\_m\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 y) t_m) z)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 0.2)
      t_2
      (if (<= t_3 20000000.0)
        (fma t_m (/ z y) t_m)
        (if (<= t_3 2e+77) (* (- t_m) (/ x y)) 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 - y) * t_m) / z;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 0.2) {
		tmp = t_2;
	} else if (t_3 <= 20000000.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_3 <= 2e+77) {
		tmp = -t_m * (x / y);
	} 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(Float64(x - y) * t_m) / z)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 0.2)
		tmp = t_2;
	elseif (t_3 <= 20000000.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_3 <= 2e+77)
		tmp = Float64(Float64(-t_m) * Float64(x / y));
	else
		tmp = t_2;
	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[(N[(x - y), $MachinePrecision] * t$95$m), $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, 0.2], t$95$2, If[LessEqual[t$95$3, 20000000.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 2e+77], N[((-t$95$m) * N[(x / y), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.20000000000000001 or 1.99999999999999997e77 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   if 0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e7

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.2%

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

   if 2e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999997e77

   1. Initial program 99.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 70.4

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

   12. Applied rewrites 70.4%

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

4. Final simplification 84.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.2:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(-t\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.2:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(-t\_m\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.2)
      (* (/ x z) t_m)
      (if (<= t_2 20000000.0)
        (fma t_m (/ z y) t_m)
        (if (<= t_2 2e+77) (* (- t_m) (/ x y)) (/ (* t_m x) z)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.2) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 20000000.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_2 <= 2e+77) {
		tmp = -t_m * (x / y);
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.2)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 20000000.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_2 <= 2e+77)
		tmp = Float64(Float64(-t_m) * Float64(x / y));
	else
		tmp = Float64(Float64(t_m * x) / z);
	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, 0.2], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 20000000.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2e+77], N[((-t$95$m) * N[(x / y), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

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

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.3

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

   5. Applied rewrites 64.3%

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

   if 0.20000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2e7

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 97.6

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 97.2%

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

   if 2e7 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999997e77

   1. Initial program 99.4%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.4

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

   4. Applied rewrites 99.4%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 70.4

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

   7. Applied rewrites 70.4%

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

   8. Taylor expanded in x around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift-*.f64 60.8

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

   10. Applied rewrites 60.8%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 70.4

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

   12. Applied rewrites 70.4%

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

   if 1.99999999999999997e77 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.1%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 71.4

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

   5. Applied rewrites 71.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.2:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20000000:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+77}:\\ \;\;\;\;\left(-t\right) \cdot \frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 80.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.001 \lor \neg \left(t\_2 \leq 2 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{-x}{y}, t\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (or (<= t_2 0.001) (not (<= t_2 2e+77)))
      (/ (* (- x y) t_m) z)
      (fma t_m (/ (- x) y) t_m)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 0.001) || !(t_2 <= 2e+77)) {
		tmp = ((x - y) * t_m) / z;
	} else {
		tmp = fma(t_m, (-x / y), t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_2 <= 0.001) || !(t_2 <= 2e+77))
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	else
		tmp = fma(t_m, Float64(Float64(-x) / y), t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 0.001], N[Not[LessEqual[t$95$2, 2e+77]], $MachinePrecision]], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], N[(t$95$m * N[((-x) / y), $MachinePrecision] + t$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1e-3 or 1.99999999999999997e77 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 1e-3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.99999999999999997e77

   1. Initial program 99.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 91.6

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

   7. Applied rewrites 91.6%

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

   8. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(t, \frac{-1 \cdot x}{y}, t\right) \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 90.9

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

   10. Applied rewrites 90.9%

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

4. Final simplification 84.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.001 \lor \neg \left(\frac{x - y}{z - y} \leq 2 \cdot 10^{+77}\right):\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{-x}{y}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 93.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.9999999999999848:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.9999999999999848)
      (* (- x y) (/ t_m (- z y)))
      (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (* (/ x (- z y)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.9999999999999848) {
		tmp = (x - y) * (t_m / (z - y));
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (x / (z - y)) * t_m;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.9999999999999848)
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.9999999999999848], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.99999999999998479

   1. Initial program 94.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. associate-/l* N/A

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

      7. lower-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 N/A

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

      10. lift--.f64 N/A

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

      11. lift--.f64 91.1

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

   4. Applied rewrites 91.1%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 92.6

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

   6. Applied rewrites 92.6%

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

   if 0.99999999999998479 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 100.0%

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

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

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 95.8%

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

Alternative 13: 70.9% accurate, 0.4× speedup

Math

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

FPCore

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

C

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

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.2)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / z);
	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, 0.2], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.3

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

   5. Applied rewrites 64.3%

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

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

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 100.0

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

   7. Applied rewrites 100.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 99.4%

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

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

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

Alternative 14: 69.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{-14} \lor \neg \left(t\_2 \leq 2\right):\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_m\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (* t_s (if (or (<= t_2 2e-14) (not (<= t_2 2.0))) (/ (* t_m x) z) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 2e-14) || !(t_2 <= 2.0)) {
		tmp = (t_m * x) / z;
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if ((t_2 <= 2d-14) .or. (.not. (t_2 <= 2.0d0))) then
        tmp = (t_m * x) / z
    else
        tmp = t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 2e-14) || !(t_2 <= 2.0)) {
		tmp = (t_m * x) / z;
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if (t_2 <= 2e-14) or not (t_2 <= 2.0):
		tmp = (t_m * x) / z
	else:
		tmp = t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_2 <= 2e-14) || !(t_2 <= 2.0))
		tmp = Float64(Float64(t_m * x) / z);
	else
		tmp = t_m;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_2 <= 2e-14) || ~((t_2 <= 2.0)))
		tmp = (t_m * x) / z;
	else
		tmp = t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 2e-14], N[Not[LessEqual[t$95$2, 2.0]], $MachinePrecision]], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision], t$95$m]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2e-14 or 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.7%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 61.2

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

   5. Applied rewrites 61.2%

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

   if 2e-14 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 94.2%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{-14} \lor \neg \left(\frac{x - y}{z - y} \leq 2\right):\\ \;\;\;\;\frac{t \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.001:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.001)
      (* (/ x z) t_m)
      (if (<= t_2 2.0) t_m (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.001) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 0.001d0) then
        tmp = (x / z) * t_m
    else if (t_2 <= 2.0d0) then
        tmp = t_m
    else
        tmp = (t_m * x) / z
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.001) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 0.001:
		tmp = (x / z) * t_m
	elif t_2 <= 2.0:
		tmp = t_m
	else:
		tmp = (t_m * x) / z
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.001)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 0.001)
		tmp = (x / z) * t_m;
	elseif (t_2 <= 2.0)
		tmp = t_m;
	else
		tmp = (t_m * x) / z;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.001], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], t$95$m, N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1e-3

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.8

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

   5. Applied rewrites 64.8%

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

   if 1e-3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 97.2%

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

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

   1. Initial program 95.8%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.5

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

   5. Applied rewrites 54.5%

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

Alternative 16: 37.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -2 \cdot 10^{-174}:\\ \;\;\;\;z \cdot \frac{t\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_m\\ \end{array} \end{array} \]

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 (<= (/ (- x y) (- z y)) -2e-174) (* z (/ t_m y)) t_m)))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (((x - y) / (z - y)) <= -2e-174) {
		tmp = z * (t_m / y);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(t_s, x, y, z, t_m)
use fmin_fmax_functions
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (((x - y) / (z - y)) <= (-2d-174)) then
        tmp = z * (t_m / y)
    else
        tmp = t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (((x - y) / (z - y)) <= -2e-174) {
		tmp = z * (t_m / y);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if ((x - y) / (z - y)) <= -2e-174:
		tmp = z * (t_m / y)
	else:
		tmp = t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (Float64(Float64(x - y) / Float64(z - y)) <= -2e-174)
		tmp = Float64(z * Float64(t_m / y));
	else
		tmp = t_m;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (((x - y) / (z - y)) <= -2e-174)
		tmp = z * (t_m / y);
	else
		tmp = t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], -2e-174], N[(z * N[(t$95$m / y), $MachinePrecision]), $MachinePrecision], t$95$m]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2e-174

   1. Initial program 94.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. div-sub N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 94.6

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

   4. Applied rewrites 94.6%

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

   5. Taylor expanded in y around inf

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

      1. sub-div N/A

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

      2. +-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-fma.f64 27.4

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

   7. Applied rewrites 27.4%

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

   8. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 6.4

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

   10. Applied rewrites 6.4%

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

       1. lift-*.f64 N/A

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

       2. lift-/.f64 N/A

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

       3. associate-/l* N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 14.5

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

   12. Applied rewrites 14.5%

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

   if -2e-174 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 97.0%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 43.7%

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

Alternative 17: 97.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

Alternative 18: 35.9% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

t\_m =     private
t\_s =     private
module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.5%

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

Developer Target 1: 97.0% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2025037 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

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

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