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

Percentage Accurate: 97.1% → 96.7%

Time: 8.9s

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

real(8) function code(x, y, z, t)
    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: 97.1% 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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.5 \cdot 10^{+48}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot t\_m}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{z - y}{t\_m}}{x - y}}\\ \end{array} \end{array} \]

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 1.5e+48)
    (/ (* (- y x) t_m) (- y z))
    (/ 1.0 (/ (/ (- z y) t_m) (- x y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 1.5e+48) {
		tmp = ((y - x) * t_m) / (y - z);
	} else {
		tmp = 1.0 / (((z - y) / t_m) / (x - y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    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 <= 1.5d+48) then
        tmp = ((y - x) * t_m) / (y - z)
    else
        tmp = 1.0d0 / (((z - y) / t_m) / (x - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 1.5e+48) {
		tmp = ((y - x) * t_m) / (y - z);
	} else {
		tmp = 1.0 / (((z - y) / t_m) / (x - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 1.5e+48:
		tmp = ((y - x) * t_m) / (y - z)
	else:
		tmp = 1.0 / (((z - y) / t_m) / (x - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 1.5e+48)
		tmp = Float64(Float64(Float64(y - x) * t_m) / Float64(y - z));
	else
		tmp = Float64(1.0 / Float64(Float64(Float64(z - y) / t_m) / Float64(x - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 1.5e+48)
		tmp = ((y - x) * t_m) / (y - z);
	else
		tmp = 1.0 / (((z - y) / t_m) / (x - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 1.5e+48], N[(N[(N[(y - x), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(N[(z - y), $MachinePrecision] / t$95$m), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.5e48

   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

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{\left(0 - \left(x - y\right)\right)} \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\left(0 - \color{blue}{\left(x - y\right)}\right) \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right) \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right) \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      12. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x\right)} \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{\color{blue}{0 - \left(z - y\right)}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{0 - \color{blue}{\left(z - y\right)}} \]

      18. sub-neg N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}} \]

      19. +-commutative N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}} \]

      20. associate--r+ N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 88.5

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

   4. Applied rewrites 88.5%

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

   if 1.5e48 < t

   1. Initial program 97.8%

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

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/r* N/A

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

      8. lower-/.f64 N/A

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

      9. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{z - y}{t}}{x - y}}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 69.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1000:\\ \;\;\;\;\frac{-x}{y} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 10^{-48}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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) t_m)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -1000.0)
      (* (/ (- x) y) t_m)
      (if (<= t_3 1e-48)
        t_2
        (if (<= t_3 5e-11)
          (* (/ (- y) z) t_m)
          (if (<= t_3 10.0) (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) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -1000.0) {
		tmp = (-x / y) * t_m;
	} else if (t_3 <= 1e-48) {
		tmp = t_2;
	} else if (t_3 <= 5e-11) {
		tmp = (-y / z) * t_m;
	} else if (t_3 <= 10.0) {
		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 / z) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -1000.0)
		tmp = Float64(Float64(Float64(-x) / y) * t_m);
	elseif (t_3 <= 1e-48)
		tmp = t_2;
	elseif (t_3 <= 5e-11)
		tmp = Float64(Float64(Float64(-y) / z) * t_m);
	elseif (t_3 <= 10.0)
		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 / z), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -1000.0], N[(N[((-x) / y), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 1e-48], t$95$2, If[LessEqual[t$95$3, 5e-11], N[(N[((-y) / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 10.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e3

   1. Initial program 91.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.9%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999997e-49 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   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 72.1

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

   5. Applied rewrites 72.1%

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

   if 9.9999999999999997e-49 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 84.7%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 97.4%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1000:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-48}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-y}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1000:\\ \;\;\;\;\frac{-x}{y} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 10^{-48}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t\_m}{-z} \cdot y\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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) t_m)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -1000.0)
      (* (/ (- x) y) t_m)
      (if (<= t_3 1e-48)
        t_2
        (if (<= t_3 5e-11)
          (* (/ t_m (- z)) y)
          (if (<= t_3 10.0) (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) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -1000.0) {
		tmp = (-x / y) * t_m;
	} else if (t_3 <= 1e-48) {
		tmp = t_2;
	} else if (t_3 <= 5e-11) {
		tmp = (t_m / -z) * y;
	} else if (t_3 <= 10.0) {
		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 / z) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -1000.0)
		tmp = Float64(Float64(Float64(-x) / y) * t_m);
	elseif (t_3 <= 1e-48)
		tmp = t_2;
	elseif (t_3 <= 5e-11)
		tmp = Float64(Float64(t_m / Float64(-z)) * y);
	elseif (t_3 <= 10.0)
		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 / z), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -1000.0], N[(N[((-x) / y), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 1e-48], t$95$2, If[LessEqual[t$95$3, 5e-11], N[(N[(t$95$m / (-z)), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t$95$3, 10.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -1e3

   1. Initial program 91.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.9%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 9.9999999999999997e-49 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   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 72.1

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

   5. Applied rewrites 72.1%

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

   if 9.9999999999999997e-49 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11

   1. Initial program 99.5%

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

      1. lift-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.5

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

   4. Applied rewrites 99.5%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 68.3

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

   7. Applied rewrites 68.3%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 66.8%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 97.4%

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

4. Final simplification 79.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1000:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{-48}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{t}{-z} \cdot y\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1000:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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 -1000.0)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 5e-11)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 10.0)
          (fma t_m (/ (- z x) 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 <= -1000.0) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 5e-11) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 10.0) {
		tmp = fma(t_m, ((z - x) / 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 <= -1000.0)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 5e-11)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 10.0)
		tmp = fma(t_m, Float64(Float64(z - x) / 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, -1000.0], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 5e-11], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 10.0], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / 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)) < -1e3

   1. Initial program 91.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11

   1. Initial program 95.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.4%

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

   if 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.4%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

Alternative 5: 93.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -1000:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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 -1000.0)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 2e-40)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0) (* (/ y (- y z)) 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 <= -1000.0) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-40) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = (x / (z - y)) * t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    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 <= (-1000.0d0)) then
        tmp = (t_m / (z - y)) * x
    else if (t_2 <= 2d-40) then
        tmp = ((x - y) / z) * t_m
    else if (t_2 <= 2.0d0) then
        tmp = (y / (y - z)) * 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 <= -1000.0) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-40) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = (y / (y - z)) * 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 <= -1000.0:
		tmp = (t_m / (z - y)) * x
	elif t_2 <= 2e-40:
		tmp = ((x - y) / z) * t_m
	elif t_2 <= 2.0:
		tmp = (y / (y - z)) * 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 <= -1000.0)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 2e-40)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(y / Float64(y - z)) * 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 <= -1000.0)
		tmp = (t_m / (z - y)) * x;
	elseif (t_2 <= 2e-40)
		tmp = ((x - y) / z) * t_m;
	elseif (t_2 <= 2.0)
		tmp = (y / (y - z)) * 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, -1000.0], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e-40], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(y / N[(y - z), $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)) < -1e3

   1. Initial program 91.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.9999999999999999e-40

   1. Initial program 95.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if 1.9999999999999999e-40 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.6

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.0

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

   10. Applied rewrites 99.0%

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

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

   1. Initial program 93.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

Alternative 6: 91.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.0001:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{y}{y - z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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 -0.0001)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 2e-40)
        (/ (* (- x y) t_m) z)
        (if (<= t_2 2.0) (* (/ y (- y z)) 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 <= -0.0001) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-40) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_2 <= 2.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = (x / (z - y)) * t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    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 <= (-0.0001d0)) then
        tmp = (t_m / (z - y)) * x
    else if (t_2 <= 2d-40) then
        tmp = ((x - y) * t_m) / z
    else if (t_2 <= 2.0d0) then
        tmp = (y / (y - z)) * 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 <= -0.0001) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 2e-40) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_2 <= 2.0) {
		tmp = (y / (y - z)) * 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 <= -0.0001:
		tmp = (t_m / (z - y)) * x
	elif t_2 <= 2e-40:
		tmp = ((x - y) * t_m) / z
	elif t_2 <= 2.0:
		tmp = (y / (y - z)) * 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 <= -0.0001)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 2e-40)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(y / Float64(y - z)) * 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 <= -0.0001)
		tmp = (t_m / (z - y)) * x;
	elseif (t_2 <= 2e-40)
		tmp = ((x - y) * t_m) / z;
	elseif (t_2 <= 2.0)
		tmp = (y / (y - z)) * 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, -0.0001], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 2e-40], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(y / N[(y - z), $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)) < -1.00000000000000005e-4

   1. Initial program 92.0%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -1.00000000000000005e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.9999999999999999e-40

   1. Initial program 95.0%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 1.9999999999999999e-40 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.6

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

   7. Applied rewrites 73.6%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.0

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

   10. Applied rewrites 99.0%

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

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

   1. Initial program 93.6%

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

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

Alternative 7: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\left(1 - \frac{x}{y}\right) \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.0001)
      t_2
      (if (<= t_3 5e-11)
        (/ (* (- x y) t_m) z)
        (if (<= t_3 10.0) (* (- 1.0 (/ x y)) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.0001) {
		tmp = t_2;
	} else if (t_3 <= 5e-11) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_3 <= 10.0) {
		tmp = (1.0 - (x / y)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-0.0001d0)) then
        tmp = t_2
    else if (t_3 <= 5d-11) then
        tmp = ((x - y) * t_m) / z
    else if (t_3 <= 10.0d0) then
        tmp = (1.0d0 - (x / y)) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.0001) {
		tmp = t_2;
	} else if (t_3 <= 5e-11) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_3 <= 10.0) {
		tmp = (1.0 - (x / y)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -0.0001:
		tmp = t_2
	elif t_3 <= 5e-11:
		tmp = ((x - y) * t_m) / z
	elif t_3 <= 10.0:
		tmp = (1.0 - (x / y)) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.0001)
		tmp = t_2;
	elseif (t_3 <= 5e-11)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_3 <= 10.0)
		tmp = Float64(Float64(1.0 - Float64(x / y)) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -0.0001)
		tmp = t_2;
	elseif (t_3 <= 5e-11)
		tmp = ((x - y) * t_m) / z;
	elseif (t_3 <= 10.0)
		tmp = (1.0 - (x / y)) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.0001], t$95$2, If[LessEqual[t$95$3, 5e-11], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, 10.0], N[(N[(1.0 - N[(x / y), $MachinePrecision]), $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)) < -1.00000000000000005e-4 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -1.00000000000000005e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11

   1. Initial program 95.3%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. distribute-neg-in N/A

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

      7. unsub-neg N/A

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

      8. remove-double-neg N/A

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

      9. lower--.f64 98.5

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

   5. Applied rewrites 98.5%

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

   7. Applied rewrites 98.5%

      \[\leadsto \left(1 - \color{blue}{\frac{x}{y}}\right) \cdot t \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\frac{y}{y - z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.0001)
      t_2
      (if (<= t_3 2e-40)
        (/ (* (- x y) t_m) z)
        (if (<= t_3 10.0) (* (/ y (- y z)) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.0001) {
		tmp = t_2;
	} else if (t_3 <= 2e-40) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_3 <= 10.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= (-0.0001d0)) then
        tmp = t_2
    else if (t_3 <= 2d-40) then
        tmp = ((x - y) * t_m) / z
    else if (t_3 <= 10.0d0) then
        tmp = (y / (y - z)) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.0001) {
		tmp = t_2;
	} else if (t_3 <= 2e-40) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_3 <= 10.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= -0.0001:
		tmp = t_2
	elif t_3 <= 2e-40:
		tmp = ((x - y) * t_m) / z
	elif t_3 <= 10.0:
		tmp = (y / (y - z)) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.0001)
		tmp = t_2;
	elseif (t_3 <= 2e-40)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_3 <= 10.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= -0.0001)
		tmp = t_2;
	elseif (t_3 <= 2e-40)
		tmp = ((x - y) * t_m) / z;
	elseif (t_3 <= 10.0)
		tmp = (y / (y - z)) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.0001], t$95$2, If[LessEqual[t$95$3, 2e-40], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, 10.0], N[(N[(y / N[(y - z), $MachinePrecision]), $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)) < -1.00000000000000005e-4 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -1.00000000000000005e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.9999999999999999e-40

   1. Initial program 95.0%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 90.6

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

   5. Applied rewrites 90.6%

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

   if 1.9999999999999999e-40 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 72.9

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.1

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

   10. Applied rewrites 98.1%

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

Alternative 9: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.0001:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.0001)
      t_2
      (if (<= t_3 5e-11)
        (/ (* (- x y) t_m) z)
        (if (<= t_3 10.0) (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 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.0001) {
		tmp = t_2;
	} else if (t_3 <= 5e-11) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_3 <= 10.0) {
		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(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.0001)
		tmp = t_2;
	elseif (t_3 <= 5e-11)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_3 <= 10.0)
		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[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.0001], t$95$2, If[LessEqual[t$95$3, 5e-11], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$3, 10.0], 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)) < -1.00000000000000005e-4 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 92.8%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 90.4

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

   5. Applied rewrites 90.4%

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

   if -1.00000000000000005e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11

   1. Initial program 95.3%

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

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 90.7

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

   5. Applied rewrites 90.7%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 97.4%

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

Alternative 10: 81.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 10^{-48}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{-y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 (* (/ t_m (- z y)) x)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 1e-48)
      t_2
      (if (<= t_3 5e-11)
        (* (/ (- y) z) t_m)
        (if (<= t_3 10.0) (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 = (t_m / (z - y)) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 1e-48) {
		tmp = t_2;
	} else if (t_3 <= 5e-11) {
		tmp = (-y / z) * t_m;
	} else if (t_3 <= 10.0) {
		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(t_m / Float64(z - y)) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 1e-48)
		tmp = t_2;
	elseif (t_3 <= 5e-11)
		tmp = Float64(Float64(Float64(-y) / z) * t_m);
	elseif (t_3 <= 10.0)
		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[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 1e-48], t$95$2, If[LessEqual[t$95$3, 5e-11], N[(N[((-y) / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 10.0], 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)) < 9.9999999999999997e-49 or 10 < (/.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

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if 9.9999999999999997e-49 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11

   1. Initial program 99.5%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 84.7%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 97.4%

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

Alternative 11: 70.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -1000:\\ \;\;\;\;\frac{-x}{y} \cdot t\_m\\ \mathbf{elif}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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) t_m)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -1000.0)
      (* (/ (- x) y) t_m)
      (if (<= t_3 5e-11) t_2 (if (<= t_3 10.0) (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) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -1000.0) {
		tmp = (-x / y) * t_m;
	} else if (t_3 <= 5e-11) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		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 / z) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -1000.0)
		tmp = Float64(Float64(Float64(-x) / y) * t_m);
	elseif (t_3 <= 5e-11)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		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 / z), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -1000.0], N[(N[((-x) / y), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 5e-11], t$95$2, If[LessEqual[t$95$3, 10.0], 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)) < -1e3

   1. Initial program 91.6%

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

   3. Taylor expanded in x around inf

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 87.7

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

   5. Applied rewrites 87.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 66.9%

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

   if -1e3 < (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.8%

      \[\frac{x - y}{z - y} \cdot t \]
   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 68.3

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

   5. Applied rewrites 68.3%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 97.4%

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

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -1000:\\ \;\;\;\;\frac{-x}{y} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-11}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 92.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot \left(x - y\right)\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\frac{y}{y - z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m (- z y)) (- x y))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 2e-40) t_2 (if (<= t_3 10.0) (* (/ y (- y z)) t_m) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * (x - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-40) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / (z - y)) * (x - y)
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-40) then
        tmp = t_2
    else if (t_3 <= 10.0d0) then
        tmp = (y / (y - z)) * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / (z - y)) * (x - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-40) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		tmp = (y / (y - z)) * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / (z - y)) * (x - y)
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-40:
		tmp = t_2
	elif t_3 <= 10.0:
		tmp = (y / (y - z)) * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / Float64(z - y)) * Float64(x - y))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-40)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		tmp = Float64(Float64(y / Float64(y - z)) * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / (z - y)) * (x - y);
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-40)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		tmp = (y / (y - z)) * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 2e-40], t$95$2, If[LessEqual[t$95$3, 10.0], N[(N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.9999999999999999e-40 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.9%

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

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 91.7

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

   4. Applied rewrites 91.7%

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

   if 1.9999999999999999e-40 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 72.9

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

   7. Applied rewrites 72.9%

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

   8. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 98.1

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

   10. Applied rewrites 98.1%

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

Alternative 13: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 5 \cdot 10^{-11}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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) t_m)) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 5e-11) t_2 (if (<= t_3 10.0) (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) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 5e-11) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		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 / z) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 5e-11)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		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 / z), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 5e-11], t$95$2, If[LessEqual[t$95$3, 10.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 5.00000000000000018e-11 or 10 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.1%

      \[\frac{x - y}{z - y} \cdot t \]
   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 61.1

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

   5. Applied rewrites 61.1%

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

   if 5.00000000000000018e-11 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   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

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \frac{t}{\frac{\color{blue}{0 - \left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      10. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z - y\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      11. sub-neg N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      12. +-commutative N/A

          \[\leadsto \frac{t}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      13. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}}{\mathsf{neg}\left(\left(x - y\right)\right)}} \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{0 - \left(x - y\right)}}} \]

      18. lift--.f64 N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x - y\right)}}} \]

      19. sub-neg N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}}} \]

      20. +-commutative N/A

          \[\leadsto \frac{t}{\frac{y - z}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}}} \]

      21. associate--r+ N/A

          \[\leadsto \frac{t}{\frac{y - z}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x}}} \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.2

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

   7. Applied rewrites 73.2%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 72.3%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 97.4%

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

Alternative 14: 68.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x}{z} \cdot t\_m\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ x z) t_m)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 2e-40) t_2 (if (<= t_3 10.0) (* 1.0 t_m) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x / z) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-40) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (x / z) * t_m
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-40) then
        tmp = t_2
    else if (t_3 <= 10.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x / z) * t_m;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-40) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x / z) * t_m
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-40:
		tmp = t_2
	elif t_3 <= 10.0:
		tmp = 1.0 * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x / z) * t_m)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-40)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x / z) * t_m;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-40)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		tmp = 1.0 * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 2e-40], t$95$2, If[LessEqual[t$95$3, 10.0], N[(1.0 * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.9999999999999999e-40 or 10 < (/.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 y around 0

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

      1. lower-/.f64 63.1

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

   5. Applied rewrites 63.1%

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

   if 1.9999999999999999e-40 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{1} \cdot t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 67.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 2 \cdot 10^{-40}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ t_m z) x)) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 2e-40) t_2 (if (<= t_3 10.0) (* 1.0 t_m) t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-40) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = (t_m / z) * x
    t_3 = (x - y) / (z - y)
    if (t_3 <= 2d-40) then
        tmp = t_2
    else if (t_3 <= 10.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (t_m / z) * x;
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 2e-40) {
		tmp = t_2;
	} else if (t_3 <= 10.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (t_m / z) * x
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 2e-40:
		tmp = t_2
	elif t_3 <= 10.0:
		tmp = 1.0 * t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(t_m / z) * x)
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 2e-40)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (t_m / z) * x;
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 2e-40)
		tmp = t_2;
	elseif (t_3 <= 10.0)
		tmp = 1.0 * t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 2e-40], t$95$2, If[LessEqual[t$95$3, 10.0], N[(1.0 * t$95$m), $MachinePrecision], t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 1.9999999999999999e-40 or 10 < (/.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

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

      1. associate-*l/ N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 78.9

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

   5. Applied rewrites 78.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 60.8%

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

   if 1.9999999999999999e-40 < (/.f64 (-.f64 x y) (-.f64 z y)) < 10

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 90.8%

      \[\leadsto \color{blue}{1} \cdot t \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 16: 97.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2.8 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(y - x\right) \cdot t\_m}{y - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot \left(x - y\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2.8e-35)
    (/ (* (- y x) t_m) (- y z))
    (* (/ t_m (- z y)) (- x y)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 2.8e-35) {
		tmp = ((y - x) * t_m) / (y - z);
	} else {
		tmp = (t_m / (z - y)) * (x - y);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    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 <= 2.8d-35) then
        tmp = ((y - x) * t_m) / (y - z)
    else
        tmp = (t_m / (z - y)) * (x - y)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 2.8e-35) {
		tmp = ((y - x) * t_m) / (y - z);
	} else {
		tmp = (t_m / (z - y)) * (x - y);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 2.8e-35:
		tmp = ((y - x) * t_m) / (y - z)
	else:
		tmp = (t_m / (z - y)) * (x - y)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 2.8e-35)
		tmp = Float64(Float64(Float64(y - x) * t_m) / Float64(y - z));
	else
		tmp = Float64(Float64(t_m / Float64(z - y)) * Float64(x - y));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 2.8e-35)
		tmp = ((y - x) * t_m) / (y - z);
	else
		tmp = (t_m / (z - y)) * (x - y);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 2.8e-35], N[(N[(N[(y - x), $MachinePrecision] * t$95$m), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.8e-35

   1. Initial program 95.3%

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

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

      3. associate-*l/ N/A

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

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

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

      7. lower-*.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto \frac{\color{blue}{\left(0 - \left(x - y\right)\right)} \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      9. lift--.f64 N/A

         \[\leadsto \frac{\left(0 - \color{blue}{\left(x - y\right)}\right) \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      10. sub-neg N/A

          \[\leadsto \frac{\left(0 - \color{blue}{\left(x + \left(\mathsf{neg}\left(y\right)\right)\right)}\right) \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      11. +-commutative N/A

          \[\leadsto \frac{\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + x\right)}\right) \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      12. associate--r+ N/A

          \[\leadsto \frac{\color{blue}{\left(\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - x\right)} \cdot t}{\mathsf{neg}\left(\left(z - y\right)\right)} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{\color{blue}{0 - \left(z - y\right)}} \]

      17. lift--.f64 N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{0 - \color{blue}{\left(z - y\right)}} \]

      18. sub-neg N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{0 - \color{blue}{\left(z + \left(\mathsf{neg}\left(y\right)\right)\right)}} \]

      19. +-commutative N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(y\right)\right) + z\right)}} \]

      20. associate--r+ N/A

          \[\leadsto \frac{\left(y - x\right) \cdot t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(y\right)\right)\right) - z}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 87.7

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

   4. Applied rewrites 87.7%

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

   if 2.8e-35 < t

   1. Initial program 98.2%

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

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

Alternative 17: 97.1% 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 = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    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 96.1%

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

Alternative 18: 34.7% accurate, 3.8× speedup

Math

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

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 (* 1.0 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 * (1.0 * t_m);
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    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 * (1.0d0 * 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 * (1.0 * 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 * (1.0 * 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(1.0 * 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 * (1.0 * 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[(1.0 * t$95$m), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 34.6%

   \[\leadsto \color{blue}{1} \cdot t \]
6. Add Preprocessing

Developer Target 1: 97.1% 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

real(8) function code(x, y, z, t)
    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 2024255 
(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))