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

Percentage Accurate: 97.2% → 97.1%

Time: 9.0s

Alternatives: 18

Speedup: N/A×

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: 97.2% 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: 97.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 1.5 \cdot 10^{+202}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot \left(x - y\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (/ (- x y) (- z y)) t_m)))
   (* t_s (if (<= t_2 1.5e+202) t_2 (* (/ 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 t_2 = ((x - y) / (z - y)) * t_m;
	double tmp;
	if (t_2 <= 1.5e+202) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_2 = ((x - y) / (z - y)) * t_m
    if (t_2 <= 1.5d+202) then
        tmp = t_2
    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 t_2 = ((x - y) / (z - y)) * t_m;
	double tmp;
	if (t_2 <= 1.5e+202) {
		tmp = t_2;
	} 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):
	t_2 = ((x - y) / (z - y)) * t_m
	tmp = 0
	if t_2 <= 1.5e+202:
		tmp = t_2
	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)
	t_2 = Float64(Float64(Float64(x - y) / Float64(z - y)) * t_m)
	tmp = 0.0
	if (t_2 <= 1.5e+202)
		tmp = t_2;
	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)
	t_2 = ((x - y) / (z - y)) * t_m;
	tmp = 0.0;
	if (t_2 <= 1.5e+202)
		tmp = t_2;
	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_] := Block[{t$95$2 = N[(N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 1.5e+202], t$95$2, 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 (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t) < 1.5000000000000001e202

   1. Initial program 98.8%

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

   if 1.5000000000000001e202 < (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t)

   1. Initial program 86.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 2: 95.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5000000000:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(-t\_m, \frac{x - z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5000000000.0)
      (* (/ x (- z y)) t_m)
      (if (<= t_2 0.4)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 200.0)
          (fma (- t_m) (/ (- x z) y) t_m)
          (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5000000000.0) {
		tmp = (x / (z - y)) * t_m;
	} else if (t_2 <= 0.4) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 200.0) {
		tmp = fma(-t_m, ((x - z) / y), t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5000000000.0)
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 200.0)
		tmp = fma(Float64(-t_m), Float64(Float64(x - z) / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5000000000.0], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[((-t$95$m) * N[(N[(x - z), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 99.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 89.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.7%

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

Alternative 3: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5000000000:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\frac{-y}{z - y} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5000000000.0)
      (* (/ x (- z y)) t_m)
      (if (<= t_2 5e-8)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 2.0) (* (/ (- y) (- z y)) t_m) (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5000000000.0) {
		tmp = (x / (z - y)) * t_m;
	} else if (t_2 <= 5e-8) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = (-y / (z - y)) * t_m;
	} else {
		tmp = (t_m * x) / (z - 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) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-5000000000.0d0)) then
        tmp = (x / (z - y)) * t_m
    else if (t_2 <= 5d-8) then
        tmp = ((x - y) / z) * t_m
    else if (t_2 <= 2.0d0) then
        tmp = (-y / (z - y)) * t_m
    else
        tmp = (t_m * x) / (z - y)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5000000000.0) {
		tmp = (x / (z - y)) * t_m;
	} else if (t_2 <= 5e-8) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = (-y / (z - y)) * t_m;
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -5000000000.0:
		tmp = (x / (z - y)) * t_m
	elif t_2 <= 5e-8:
		tmp = ((x - y) / z) * t_m
	elif t_2 <= 2.0:
		tmp = (-y / (z - y)) * t_m
	else:
		tmp = (t_m * x) / (z - y)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5000000000.0)
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	elseif (t_2 <= 5e-8)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = Float64(Float64(Float64(-y) / Float64(z - y)) * t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -5000000000.0)
		tmp = (x / (z - y)) * t_m;
	elseif (t_2 <= 5e-8)
		tmp = ((x - y) / z) * t_m;
	elseif (t_2 <= 2.0)
		tmp = (-y / (z - y)) * t_m;
	else
		tmp = (t_m * x) / (z - y);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5000000000.0], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 5e-8], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[((-y) / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 99.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 99.1

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

   5. Applied rewrites 99.1%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 97.1

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

   5. Applied rewrites 97.1%

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

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

   1. Initial program 89.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 96.3%

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

Alternative 4: 94.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5000000000:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{x - y}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(-t\_m, \frac{x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5000000000.0)
      (* (/ x (- z y)) t_m)
      (if (<= t_2 0.4)
        (* (/ (- x y) z) t_m)
        (if (<= t_2 200.0)
          (fma (- t_m) (/ x y) t_m)
          (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5000000000.0) {
		tmp = (x / (z - y)) * t_m;
	} else if (t_2 <= 0.4) {
		tmp = ((x - y) / z) * t_m;
	} else if (t_2 <= 200.0) {
		tmp = fma(-t_m, (x / y), t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5000000000.0)
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(Float64(x - y) / z) * t_m);
	elseif (t_2 <= 200.0)
		tmp = fma(Float64(-t_m), Float64(x / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5000000000.0], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[((-t$95$m) * N[(x / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 96.3

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

   5. Applied rewrites 96.3%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 99.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 96.7

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

   5. Applied rewrites 96.7%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.9%

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

   if 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 89.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.7%

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

Alternative 5: 93.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{x}{z - y} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(-t\_m, \frac{x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -0.1)
      (* (/ x (- z y)) t_m)
      (if (<= t_2 0.4)
        (/ (* (- x y) t_m) z)
        (if (<= t_2 200.0)
          (fma (- t_m) (/ x y) t_m)
          (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (x / (z - y)) * t_m;
	} else if (t_2 <= 0.4) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_2 <= 200.0) {
		tmp = fma(-t_m, (x / y), t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(Float64(x / Float64(z - y)) * t_m);
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_2 <= 200.0)
		tmp = fma(Float64(-t_m), Float64(x / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -0.1], N[(N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[((-t$95$m) * N[(x / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -0.10000000000000001

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 96.4

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

   5. Applied rewrites 96.4%

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

   if -0.10000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      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 90.5

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

   5. Applied rewrites 90.5%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.9%

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

   if 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 89.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.7%

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

Alternative 6: 92.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(-t\_m, \frac{x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -0.1)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 0.4)
        (/ (* (- x y) t_m) z)
        (if (<= t_2 200.0)
          (fma (- t_m) (/ x y) t_m)
          (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 0.4) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_2 <= 200.0) {
		tmp = fma(-t_m, (x / y), t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_2 <= 200.0)
		tmp = fma(Float64(-t_m), Float64(x / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -0.1], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[((-t$95$m) * N[(x / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -0.10000000000000001

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

   if -0.10000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      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 90.5

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

   5. Applied rewrites 90.5%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 97.9%

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

   if 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 89.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   7. Applied rewrites 97.7%

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

Alternative 7: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -0.1:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t\_m}{y}, x, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -0.1)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 0.4)
        (/ (* (- x y) t_m) z)
        (if (<= t_2 2.0) (fma (/ (- t_m) y) x t_m) (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -0.1) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 0.4) {
		tmp = ((x - y) * t_m) / z;
	} else if (t_2 <= 2.0) {
		tmp = fma((-t_m / y), x, t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -0.1)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(Float64(x - y) * t_m) / z);
	elseif (t_2 <= 2.0)
		tmp = fma(Float64(Float64(-t_m) / y), x, t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -0.1], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(N[(x - y), $MachinePrecision] * t$95$m), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[((-t$95$m) / y), $MachinePrecision] * x + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -0.10000000000000001

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 87.3

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

   5. Applied rewrites 87.3%

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

   if -0.10000000000000001 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      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 90.5

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

   5. Applied rewrites 90.5%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 97.1%

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

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

   1. Initial program 89.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 96.3%

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

4. Final simplification 93.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -0.1:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\frac{\left(x - y\right) \cdot t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5000000000:\\ \;\;\;\;\frac{t\_m}{z - y} \cdot x\\ \mathbf{elif}\;t\_2 \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_m}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t\_m}{y}, x, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5000000000.0)
      (* (/ t_m (- z y)) x)
      (if (<= t_2 8e-5)
        (* (/ t_m z) (- x y))
        (if (<= t_2 2.0) (fma (/ (- t_m) y) x t_m) (/ (* t_m x) (- z y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5000000000.0) {
		tmp = (t_m / (z - y)) * x;
	} else if (t_2 <= 8e-5) {
		tmp = (t_m / z) * (x - y);
	} else if (t_2 <= 2.0) {
		tmp = fma((-t_m / y), x, t_m);
	} else {
		tmp = (t_m * x) / (z - y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5000000000.0)
		tmp = Float64(Float64(t_m / Float64(z - y)) * x);
	elseif (t_2 <= 8e-5)
		tmp = Float64(Float64(t_m / z) * Float64(x - y));
	elseif (t_2 <= 2.0)
		tmp = fma(Float64(Float64(-t_m) / y), x, t_m);
	else
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5000000000.0], N[(N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[t$95$2, 8e-5], N[(N[(t$95$m / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[((-t$95$m) / y), $MachinePrecision] * x + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9

   1. Initial program 97.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 86.7

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

   5. Applied rewrites 86.7%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 8.00000000000000065e-5

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.2%

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

   if 8.00000000000000065e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in 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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 96.7

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

   5. Applied rewrites 96.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 95.2%

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

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

   1. Initial program 89.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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 95.6

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

   5. Applied rewrites 95.6%

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

   7. Applied rewrites 96.3%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5000000000:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 91.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y} \cdot x\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -5000000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_m}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;t\_3 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t\_m}{y}, x, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

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 -5000000000.0)
      t_2
      (if (<= t_3 8e-5)
        (* (/ t_m z) (- x y))
        (if (<= t_3 200.0) (fma (/ (- t_m) y) x 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 <= -5000000000.0) {
		tmp = t_2;
	} else if (t_3 <= 8e-5) {
		tmp = (t_m / z) * (x - y);
	} else if (t_3 <= 200.0) {
		tmp = fma((-t_m / y), x, 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 <= -5000000000.0)
		tmp = t_2;
	elseif (t_3 <= 8e-5)
		tmp = Float64(Float64(t_m / z) * Float64(x - y));
	elseif (t_3 <= 200.0)
		tmp = fma(Float64(Float64(-t_m) / y), x, 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, -5000000000.0], t$95$2, If[LessEqual[t$95$3, 8e-5], N[(N[(t$95$m / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 200.0], N[(N[((-t$95$m) / y), $MachinePrecision] * x + t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5e9 or 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.1%

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

   3. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{t \cdot x}{z - y}} \]
   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 92.5

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

   5. Applied rewrites 92.5%

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

   if -5e9 < (/.f64 (-.f64 x y) (-.f64 z y)) < 8.00000000000000065e-5

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      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 91.1

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

   5. Applied rewrites 91.1%

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

   7. Applied rewrites 91.2%

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

   if 8.00000000000000065e-5 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 96.8

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

   5. Applied rewrites 96.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 94.3%

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

4. Final simplification 92.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5000000000:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{y}, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - y} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 70.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(-t\_m\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t\_m, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(-t\_m\right) \cdot x}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5e+43)
      (* (- t_m) (/ x y))
      (if (<= t_2 0.4)
        (* (/ x z) t_m)
        (if (<= t_2 2.0) (fma (/ z y) t_m t_m) (/ (* (- 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 t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+43) {
		tmp = -t_m * (x / y);
	} else if (t_2 <= 0.4) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 2.0) {
		tmp = fma((z / y), t_m, t_m);
	} else {
		tmp = (-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)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5e+43)
		tmp = Float64(Float64(-t_m) * Float64(x / y));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 2.0)
		tmp = fma(Float64(z / y), t_m, t_m);
	else
		tmp = Float64(Float64(Float64(-t_m) * x) / y);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e+43], N[((-t$95$m) * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(N[(z / y), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision], N[(N[((-t$95$m) * x), $MachinePrecision] / y), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5.0000000000000004e43

   1. Initial program 96.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.7%

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

   if -5.0000000000000004e43 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 96.0%

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

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

   1. Initial program 89.6%

      \[\frac{x - y}{z - y} \cdot t \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 59.2

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

   5. Applied rewrites 59.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 57.6%

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

   10. Applied rewrites 61.7%

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

Alternative 11: 70.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(-t\_m\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t\_m, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{z} \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5e+43)
      (* (- t_m) (/ x y))
      (if (<= t_2 0.4)
        (* (/ x z) t_m)
        (if (<= t_2 200.0) (fma (/ z y) t_m t_m) (* (/ t_m z) x)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+43) {
		tmp = -t_m * (x / y);
	} else if (t_2 <= 0.4) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 200.0) {
		tmp = fma((z / y), t_m, t_m);
	} else {
		tmp = (t_m / z) * x;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5e+43)
		tmp = Float64(Float64(-t_m) * Float64(x / y));
	elseif (t_2 <= 0.4)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 200.0)
		tmp = fma(Float64(z / y), t_m, t_m);
	else
		tmp = Float64(Float64(t_m / z) * x);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e+43], N[((-t$95$m) * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[(N[(z / y), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision], N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5.0000000000000004e43

   1. Initial program 96.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.7%

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

   if -5.0000000000000004e43 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 99.4%

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

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.7

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

   5. Applied rewrites 64.7%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.1%

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

   if 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 89.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.4%

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

Alternative 12: 37.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y} \cdot t\_m\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0 \lor \neg \left(t\_2 \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;z \cdot \frac{t\_m}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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_m)))
   (*
    t_s
    (if (or (<= t_2 0.0) (not (<= t_2 5e+306))) (* z (/ t_m y)) (* 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) {
	double t_2 = ((x - y) / (z - y)) * t_m;
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 5e+306)) {
		tmp = z * (t_m / y);
	} else {
		tmp = 1.0 * 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)) * t_m
    if ((t_2 <= 0.0d0) .or. (.not. (t_2 <= 5d+306))) then
        tmp = z * (t_m / y)
    else
        tmp = 1.0d0 * 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)) * t_m;
	double tmp;
	if ((t_2 <= 0.0) || !(t_2 <= 5e+306)) {
		tmp = z * (t_m / y);
	} else {
		tmp = 1.0 * 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)) * t_m
	tmp = 0
	if (t_2 <= 0.0) or not (t_2 <= 5e+306):
		tmp = z * (t_m / y)
	else:
		tmp = 1.0 * 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(Float64(x - y) / Float64(z - y)) * t_m)
	tmp = 0.0
	if ((t_2 <= 0.0) || !(t_2 <= 5e+306))
		tmp = Float64(z * Float64(t_m / y));
	else
		tmp = Float64(1.0 * 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)) * t_m;
	tmp = 0.0;
	if ((t_2 <= 0.0) || ~((t_2 <= 5e+306)))
		tmp = z * (t_m / y);
	else
		tmp = 1.0 * 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[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision] * t$95$m), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 0.0], N[Not[LessEqual[t$95$2, 5e+306]], $MachinePrecision]], N[(z * N[(t$95$m / y), $MachinePrecision]), $MachinePrecision], N[(1.0 * t$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t) < -0.0 or 4.99999999999999993e306 < (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t)

   1. Initial program 96.1%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 55.0

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

   5. Applied rewrites 55.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 5.5%

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

   10. Applied rewrites 10.2%

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

   if -0.0 < (*.f64 (/.f64 (-.f64 x y) (-.f64 z y)) t) < 4.99999999999999993e306

   1. Initial program 99.8%

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

   3. Taylor expanded in y around inf

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

   5. Applied rewrites 39.3%

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

4. Final simplification 21.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \cdot t \leq 0 \lor \neg \left(\frac{x - y}{z - y} \cdot t \leq 5 \cdot 10^{+306}\right):\\ \;\;\;\;z \cdot \frac{t}{y}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 79.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(-t\_m\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_2 \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t\_m}{z} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t\_m}{y}, x, t\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5e+43)
      (* (- t_m) (/ x y))
      (if (<= t_2 8e-5) (* (/ t_m z) (- x y)) (fma (/ (- t_m) y) x t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+43) {
		tmp = -t_m * (x / y);
	} else if (t_2 <= 8e-5) {
		tmp = (t_m / z) * (x - y);
	} else {
		tmp = fma((-t_m / y), x, t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5e+43)
		tmp = Float64(Float64(-t_m) * Float64(x / y));
	elseif (t_2 <= 8e-5)
		tmp = Float64(Float64(t_m / z) * Float64(x - y));
	else
		tmp = fma(Float64(Float64(-t_m) / y), x, t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e+43], N[((-t$95$m) * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 8e-5], N[(N[(t$95$m / z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision], N[(N[((-t$95$m) / y), $MachinePrecision] * x + t$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5.0000000000000004e43

   1. Initial program 96.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.7%

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

   if -5.0000000000000004e43 < (/.f64 (-.f64 x y) (-.f64 z y)) < 8.00000000000000065e-5

   1. Initial program 99.4%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      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 87.3

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

   5. Applied rewrites 87.3%

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

   7. Applied rewrites 88.5%

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

   if 8.00000000000000065e-5 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.9%

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

4. Final simplification 82.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(-t\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 8 \cdot 10^{-5}:\\ \;\;\;\;\frac{t}{z} \cdot \left(x - y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{y}, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(-t\_m\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t\_m}{y}, x, t\_m\right)\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -5e+43)
      (* (- t_m) (/ x y))
      (if (<= t_2 5e-8) (* (/ x z) t_m) (fma (/ (- t_m) y) x t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -5e+43) {
		tmp = -t_m * (x / y);
	} else if (t_2 <= 5e-8) {
		tmp = (x / z) * t_m;
	} else {
		tmp = fma((-t_m / y), x, t_m);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -5e+43)
		tmp = Float64(Float64(-t_m) * Float64(x / y));
	elseif (t_2 <= 5e-8)
		tmp = Float64(Float64(x / z) * t_m);
	else
		tmp = fma(Float64(Float64(-t_m) / y), x, t_m);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -5e+43], N[((-t$95$m) * N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-8], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], N[(N[((-t$95$m) / y), $MachinePrecision] * x + t$95$m), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -5.0000000000000004e43

   1. Initial program 96.9%

      \[\frac{x - y}{z - y} \cdot t \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 62.7%

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

   if -5.0000000000000004e43 < (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 99.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 66.8

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

   5. Applied rewrites 66.8%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 83.2

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

   5. Applied rewrites 83.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 82.4%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -5 \cdot 10^{+43}:\\ \;\;\;\;\left(-t\right) \cdot \frac{x}{y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-t}{y}, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 70.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{y}, t\_m, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{z} \cdot x\\ \end{array} \end{array} \end{array} \]

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 0.4)
      (* (/ x z) t_m)
      (if (<= t_2 200.0) (fma (/ z y) t_m t_m) (* (/ t_m z) x))))))

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 <= 0.4) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 200.0) {
		tmp = fma((z / y), t_m, t_m);
	} else {
		tmp = (t_m / z) * x;
	}
	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 <= 0.4)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 200.0)
		tmp = fma(Float64(z / y), t_m, t_m);
	else
		tmp = Float64(Float64(t_m / z) * x);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[(N[(z / y), $MachinePrecision] * t$95$m + t$95$m), $MachinePrecision], N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 98.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 59.1

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

   5. Applied rewrites 59.1%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\left(t + -1 \cdot \frac{t \cdot x}{y}\right) - -1 \cdot \frac{t \cdot z}{y}} \]
   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-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-neg.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 95.1%

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

   if 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 89.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.4%

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

Alternative 16: 68.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-8} \lor \neg \left(t\_2 \leq 200\right):\\ \;\;\;\;\frac{t\_m}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\_m\\ \end{array} \end{array} \end{array} \]

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 (or (<= t_2 5e-8) (not (<= t_2 200.0))) (* (/ t_m z) x) (* 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) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 5e-8) || !(t_2 <= 200.0)) {
		tmp = (t_m / z) * x;
	} else {
		tmp = 1.0 * 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 <= 5d-8) .or. (.not. (t_2 <= 200.0d0))) then
        tmp = (t_m / z) * x
    else
        tmp = 1.0d0 * t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if ((t_2 <= 5e-8) || !(t_2 <= 200.0)) {
		tmp = (t_m / z) * x;
	} else {
		tmp = 1.0 * t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if (t_2 <= 5e-8) or not (t_2 <= 200.0):
		tmp = (t_m / z) * x
	else:
		tmp = 1.0 * t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if ((t_2 <= 5e-8) || !(t_2 <= 200.0))
		tmp = Float64(Float64(t_m / z) * x);
	else
		tmp = Float64(1.0 * t_m);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if ((t_2 <= 5e-8) || ~((t_2 <= 200.0)))
		tmp = (t_m / z) * x;
	else
		tmp = 1.0 * t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[Or[LessEqual[t$95$2, 5e-8], N[Not[LessEqual[t$95$2, 200.0]], $MachinePrecision]], N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * t$95$m), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8 or 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 96.1%

      \[\frac{x - y}{z - y} \cdot t \]
   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 78.1

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

   5. Applied rewrites 78.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.2%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   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 91.5%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 5 \cdot 10^{-8} \lor \neg \left(\frac{x - y}{z - y} \leq 200\right):\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 69.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{x}{z} \cdot t\_m\\ \mathbf{elif}\;t\_2 \leq 200:\\ \;\;\;\;1 \cdot t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{z} \cdot x\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 5e-8)
      (* (/ x z) t_m)
      (if (<= t_2 200.0) (* 1.0 t_m) (* (/ t_m z) x))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 5e-8) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 200.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = (t_m / z) * x;
	}
	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 <= 5d-8) then
        tmp = (x / z) * t_m
    else if (t_2 <= 200.0d0) then
        tmp = 1.0d0 * t_m
    else
        tmp = (t_m / z) * x
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 5e-8) {
		tmp = (x / z) * t_m;
	} else if (t_2 <= 200.0) {
		tmp = 1.0 * t_m;
	} else {
		tmp = (t_m / z) * x;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 5e-8:
		tmp = (x / z) * t_m
	elif t_2 <= 200.0:
		tmp = 1.0 * t_m
	else:
		tmp = (t_m / z) * x
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 5e-8)
		tmp = Float64(Float64(x / z) * t_m);
	elseif (t_2 <= 200.0)
		tmp = Float64(1.0 * t_m);
	else
		tmp = Float64(Float64(t_m / z) * x);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 5e-8)
		tmp = (x / z) * t_m;
	elseif (t_2 <= 200.0)
		tmp = 1.0 * t_m;
	else
		tmp = (t_m / z) * x;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 5e-8], N[(N[(x / z), $MachinePrecision] * t$95$m), $MachinePrecision], If[LessEqual[t$95$2, 200.0], N[(1.0 * t$95$m), $MachinePrecision], N[(N[(t$95$m / z), $MachinePrecision] * x), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 4.9999999999999998e-8

   1. Initial program 98.7%

      \[\frac{x - y}{z - y} \cdot t \]
   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 60.5

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

   5. Applied rewrites 60.5%

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

   if 4.9999999999999998e-8 < (/.f64 (-.f64 x y) (-.f64 z y)) < 200

   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 91.5%

      \[\leadsto \color{blue}{1} \cdot t \]

   if 200 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 89.4%

      \[\frac{x - y}{z - y} \cdot t \]
   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 97.6

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

   5. Applied rewrites 97.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 61.4%

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

Alternative 18: 34.8% 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 97.5%

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

3. Taylor expanded in y around inf

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

5. Applied rewrites 35.5%

   \[\leadsto \color{blue}{1} \cdot t \]
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 2024338 
(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))