Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A

Percentage Accurate: 88.5% → 99.8%

Time: 7.2s

Alternatives: 8

Speedup: 0.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{1 - \frac{y}{z}} \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (/ (+ x y) (- 1.0 (/ y z))))

C

double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (x + y) / (1.0d0 - (y / z))
end function

Java

public static double code(double x, double y, double z) {
	return (x + y) / (1.0 - (y / z));
}

Python

def code(x, y, z):
	return (x + y) / (1.0 - (y / z))

Julia

function code(x, y, z)
	return Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x + y) / (1.0 - (y / z));
end

Wolfram

code[x_, y_, z_] := N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;-\left(\frac{\left(\left(\frac{\left(z + x\right) \cdot z}{y} + x\right) + z\right) \cdot z}{y} + z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-297)
     (* (/ z (- z y)) (+ y x))
     (if (<= t_0 0.0)
       (- (+ (/ (* (+ (+ (/ (* (+ z x) z) y) x) z) z) y) z))
       t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = (z / (z - y)) * (y + x);
	} else if (t_0 <= 0.0) {
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    if (t_0 <= (-5d-297)) then
        tmp = (z / (z - y)) * (y + x)
    else if (t_0 <= 0.0d0) then
        tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = (z / (z - y)) * (y + x);
	} else if (t_0 <= 0.0) {
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -5e-297:
		tmp = (z / (z - y)) * (y + x)
	elif t_0 <= 0.0:
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -5e-297)
		tmp = Float64(Float64(z / Float64(z - y)) * Float64(y + x));
	elseif (t_0 <= 0.0)
		tmp = Float64(-Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(z + x) * z) / y) + x) + z) * z) / y) + z));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -5e-297)
		tmp = (z / (z - y)) * (y + x);
	elseif (t_0 <= 0.0)
		tmp = -((((((((z + x) * z) / y) + x) + z) * z) / y) + z);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-297], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], (-N[(N[(N[(N[(N[(N[(N[(N[(z + x), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision] + x), $MachinePrecision] + z), $MachinePrecision] * z), $MachinePrecision] / y), $MachinePrecision] + z), $MachinePrecision]), t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-297

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -5e-297 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around -inf

      \[\leadsto \color{blue}{-1 \cdot z + -1 \cdot \frac{\left(x \cdot z + \frac{z \cdot \left(x \cdot z - -1 \cdot {z}^{2}\right)}{y}\right) - -1 \cdot {z}^{2}}{y}} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(x \cdot z + \frac{z \cdot \left(x \cdot z - -1 \cdot {z}^{2}\right)}{y}\right) - -1 \cdot {z}^{2}}{y} + -1 \cdot z} \]

      2. mul-1-neg N/A

         \[\leadsto -1 \cdot \frac{\left(x \cdot z + \frac{z \cdot \left(x \cdot z - -1 \cdot {z}^{2}\right)}{y}\right) - -1 \cdot {z}^{2}}{y} + \color{blue}{\left(\mathsf{neg}\left(z\right)\right)} \]

      3. unsub-neg N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(x \cdot z + \frac{z \cdot \left(x \cdot z - -1 \cdot {z}^{2}\right)}{y}\right) - -1 \cdot {z}^{2}}{y} - z} \]

      4. lower--.f64 N/A

         \[\leadsto \color{blue}{-1 \cdot \frac{\left(x \cdot z + \frac{z \cdot \left(x \cdot z - -1 \cdot {z}^{2}\right)}{y}\right) - -1 \cdot {z}^{2}}{y} - z} \]

   5. Applied rewrites 99.9%

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

   if -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;-\left(\frac{\left(\left(\frac{\left(z + x\right) \cdot z}{y} + x\right) + z\right) \cdot z}{y} + z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{z \cdot \left(y + x\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -5e-297)
     (* (/ z (- z y)) (+ y x))
     (if (<= t_0 0.0) (/ (* z (+ y x)) (- z y)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = (z / (z - y)) * (y + x);
	} else if (t_0 <= 0.0) {
		tmp = (z * (y + x)) / (z - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    if (t_0 <= (-5d-297)) then
        tmp = (z / (z - y)) * (y + x)
    else if (t_0 <= 0.0d0) then
        tmp = (z * (y + x)) / (z - y)
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = (z / (z - y)) * (y + x);
	} else if (t_0 <= 0.0) {
		tmp = (z * (y + x)) / (z - y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -5e-297:
		tmp = (z / (z - y)) * (y + x)
	elif t_0 <= 0.0:
		tmp = (z * (y + x)) / (z - y)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	tmp = 0.0
	if (t_0 <= -5e-297)
		tmp = Float64(Float64(z / Float64(z - y)) * Float64(y + x));
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(z * Float64(y + x)) / Float64(z - y));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	tmp = 0.0;
	if (t_0 <= -5e-297)
		tmp = (z / (z - y)) * (y + x);
	elseif (t_0 <= 0.0)
		tmp = (z * (y + x)) / (z - y);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-297], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$0, 0.0], N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-297

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.9

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   if -5e-297 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 5.8

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

   4. Applied rewrites 5.8%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 10.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 10.7

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

   6. Applied rewrites 10.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift--.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{z \cdot \left(y + x\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ t_1 := \frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-297}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{z \cdot \left(y + x\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))) (t_1 (* (/ z (- z y)) (+ y x))))
   (if (<= t_0 -5e-297) t_1 (if (<= t_0 0.0) (/ (* z (+ y x)) (- z y)) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (y + x);
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (z * (y + x)) / (z - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    t_1 = (z / (z - y)) * (y + x)
    if (t_0 <= (-5d-297)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = (z * (y + x)) / (z - y)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (y + x);
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (z * (y + x)) / (z - y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	t_1 = (z / (z - y)) * (y + x)
	tmp = 0
	if t_0 <= -5e-297:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (z * (y + x)) / (z - y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(z / Float64(z - y)) * Float64(y + x))
	tmp = 0.0
	if (t_0 <= -5e-297)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(z * Float64(y + x)) / Float64(z - y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	t_1 = (z / (z - y)) * (y + x);
	tmp = 0.0;
	if (t_0 <= -5e-297)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (z * (y + x)) / (z - y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-297], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(z * N[(y + x), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-297 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -5e-297 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 5.8

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

   4. Applied rewrites 5.8%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 10.7

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 10.7

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

   6. Applied rewrites 10.7%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. associate-*l/ N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. lift-+.f64 N/A

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

      9. +-commutative N/A

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

      10. lower-+.f64 N/A

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

      11. lift--.f64 99.8

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

   8. Applied rewrites 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{z \cdot \left(y + x\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ t_1 := \frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{if}\;t\_0 \leq -5 \cdot 10^{-297}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-x}{y} \cdot z - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))) (t_1 (* (/ z (- z y)) (+ y x))))
   (if (<= t_0 -5e-297) t_1 (if (<= t_0 0.0) (- (* (/ (- x) y) z) z) t_1))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (y + x);
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((-x / y) * z) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = (y + x) / (1.0d0 - (y / z))
    t_1 = (z / (z - y)) * (y + x)
    if (t_0 <= (-5d-297)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = ((-x / y) * z) - z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double t_1 = (z / (z - y)) * (y + x);
	double tmp;
	if (t_0 <= -5e-297) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = ((-x / y) * z) - z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	t_1 = (z / (z - y)) * (y + x)
	tmp = 0
	if t_0 <= -5e-297:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = ((-x / y) * z) - z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + x) / Float64(1.0 - Float64(y / z)))
	t_1 = Float64(Float64(z / Float64(z - y)) * Float64(y + x))
	tmp = 0.0
	if (t_0 <= -5e-297)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(-x) / y) * z) - z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + x) / (1.0 - (y / z));
	t_1 = (z / (z - y)) * (y + x);
	tmp = 0.0;
	if (t_0 <= -5e-297)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = ((-x / y) * z) - z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + x), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, -5e-297], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision] - z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -5e-297 or -0.0 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z)))

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 99.8

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -5e-297 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -0.0

   1. Initial program 5.8%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 99.7

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

   5. Applied rewrites 99.7%

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

   7. Applied rewrites 99.7%

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

   9. Applied rewrites 99.7%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -5 \cdot 10^{-297}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{-x}{y} \cdot z - z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 74.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{if}\;y \leq -1 \cdot 10^{-20}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-14}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+114}:\\ \;\;\;\;\frac{z}{z - y} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- -1.0 (/ x y)) z)))
   (if (<= y -1e-20)
     t_0
     (if (<= y 3.6e-14)
       (+ y x)
       (if (<= y 2.5e+114) (* (/ z (- z y)) y) t_0)))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 - (x / y)) * z;
	double tmp;
	if (y <= -1e-20) {
		tmp = t_0;
	} else if (y <= 3.6e-14) {
		tmp = y + x;
	} else if (y <= 2.5e+114) {
		tmp = (z / (z - y)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((-1.0d0) - (x / y)) * z
    if (y <= (-1d-20)) then
        tmp = t_0
    else if (y <= 3.6d-14) then
        tmp = y + x
    else if (y <= 2.5d+114) then
        tmp = (z / (z - y)) * y
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (-1.0 - (x / y)) * z;
	double tmp;
	if (y <= -1e-20) {
		tmp = t_0;
	} else if (y <= 3.6e-14) {
		tmp = y + x;
	} else if (y <= 2.5e+114) {
		tmp = (z / (z - y)) * y;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 - (x / y)) * z
	tmp = 0
	if y <= -1e-20:
		tmp = t_0
	elif y <= 3.6e-14:
		tmp = y + x
	elif y <= 2.5e+114:
		tmp = (z / (z - y)) * y
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(-1.0 - Float64(x / y)) * z)
	tmp = 0.0
	if (y <= -1e-20)
		tmp = t_0;
	elseif (y <= 3.6e-14)
		tmp = Float64(y + x);
	elseif (y <= 2.5e+114)
		tmp = Float64(Float64(z / Float64(z - y)) * y);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (-1.0 - (x / y)) * z;
	tmp = 0.0;
	if (y <= -1e-20)
		tmp = t_0;
	elseif (y <= 3.6e-14)
		tmp = y + x;
	elseif (y <= 2.5e+114)
		tmp = (z / (z - y)) * y;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -1e-20], t$95$0, If[LessEqual[y, 3.6e-14], N[(y + x), $MachinePrecision], If[LessEqual[y, 2.5e+114], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$0]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.99999999999999945e-21 or 2.5e114 < y

   1. Initial program 75.7%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. associate-*l* N/A

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

      4. lower-*.f64 N/A

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

      5. associate-*r/ N/A

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

      6. +-commutative N/A

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

      7. distribute-lft-in N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. div-sub N/A

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

      11. associate-*l/ N/A

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

      12. metadata-eval N/A

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

      13. distribute-neg-frac N/A

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

      14. distribute-lft-neg-out N/A

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

      15. lft-mult-inverse N/A

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

      16. metadata-eval N/A

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

      17. lower--.f64 N/A

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

      18. lower-/.f64 81.5

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

   5. Applied rewrites 81.5%

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

   if -9.99999999999999945e-21 < y < 3.5999999999999998e-14

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 80.1

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{y + x} \]

   if 3.5999999999999998e-14 < y < 2.5e114

   1. Initial program 91.6%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 91.5

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

   4. Applied rewrites 91.5%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 91.5

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 91.5

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

   6. Applied rewrites 91.5%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 83.1

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

   9. Applied rewrites 83.1%

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

Alternative 6: 68.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{z}{z - y} \cdot y\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{-24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.6 \cdot 10^{-14}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 4.1 \cdot 10^{+198}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ z (- z y)) y)))
   (if (<= y -1.1e-24)
     t_0
     (if (<= y 3.6e-14) (+ y x) (if (<= y 4.1e+198) t_0 (- z))))))

C

double code(double x, double y, double z) {
	double t_0 = (z / (z - y)) * y;
	double tmp;
	if (y <= -1.1e-24) {
		tmp = t_0;
	} else if (y <= 3.6e-14) {
		tmp = y + x;
	} else if (y <= 4.1e+198) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (z / (z - y)) * y
    if (y <= (-1.1d-24)) then
        tmp = t_0
    else if (y <= 3.6d-14) then
        tmp = y + x
    else if (y <= 4.1d+198) then
        tmp = t_0
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = (z / (z - y)) * y;
	double tmp;
	if (y <= -1.1e-24) {
		tmp = t_0;
	} else if (y <= 3.6e-14) {
		tmp = y + x;
	} else if (y <= 4.1e+198) {
		tmp = t_0;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z / (z - y)) * y
	tmp = 0
	if y <= -1.1e-24:
		tmp = t_0
	elif y <= 3.6e-14:
		tmp = y + x
	elif y <= 4.1e+198:
		tmp = t_0
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z / Float64(z - y)) * y)
	tmp = 0.0
	if (y <= -1.1e-24)
		tmp = t_0;
	elseif (y <= 3.6e-14)
		tmp = Float64(y + x);
	elseif (y <= 4.1e+198)
		tmp = t_0;
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z / (z - y)) * y;
	tmp = 0.0;
	if (y <= -1.1e-24)
		tmp = t_0;
	elseif (y <= 3.6e-14)
		tmp = y + x;
	elseif (y <= 4.1e+198)
		tmp = t_0;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1.1e-24], t$95$0, If[LessEqual[y, 3.6e-14], N[(y + x), $MachinePrecision], If[LessEqual[y, 4.1e+198], t$95$0, (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.10000000000000001e-24 or 3.5999999999999998e-14 < y < 4.1000000000000002e198

   1. Initial program 83.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. distribute-frac-neg2 N/A

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

      10. distribute-neg-frac N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 83.3

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

   4. Applied rewrites 83.3%

      \[\leadsto \frac{x + y}{\color{blue}{\mathsf{fma}\left(\frac{-1}{z}, y, 1\right)}} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift-fma.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. associate-*l/ N/A

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

      9. div-inv N/A

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

      10. neg-mul-1 N/A

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

      11. cancel-sign-sub-inv N/A

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

      12. div-inv N/A

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

      13. *-inverses N/A

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

      14. div-sub N/A

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

      15. lift--.f64 N/A

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

      16. clear-num N/A

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

      17. lower-/.f64 84.2

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

      18. lift-+.f64 N/A

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

      19. +-commutative N/A

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

      20. lift-+.f64 84.2

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

   6. Applied rewrites 84.2%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 68.0

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

   9. Applied rewrites 68.0%

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

   if -1.10000000000000001e-24 < y < 3.5999999999999998e-14

   1. Initial program 99.9%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 80.1

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 80.1%

      \[\leadsto \color{blue}{y + x} \]

   if 4.1000000000000002e198 < y

   1. Initial program 50.4%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 86.6

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 86.6%

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

Alternative 7: 67.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.3 \cdot 10^{-15}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{+96}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.3e-15) (- z) (if (<= y 2.6e+96) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e-15) {
		tmp = -z;
	} else if (y <= 2.6e+96) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: tmp
    if (y <= (-1.3d-15)) then
        tmp = -z
    else if (y <= 2.6d+96) then
        tmp = y + x
    else
        tmp = -z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.3e-15) {
		tmp = -z;
	} else if (y <= 2.6e+96) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.3e-15:
		tmp = -z
	elif y <= 2.6e+96:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.3e-15)
		tmp = Float64(-z);
	elseif (y <= 2.6e+96)
		tmp = Float64(y + x);
	else
		tmp = Float64(-z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.3e-15)
		tmp = -z;
	elseif (y <= 2.6e+96)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.3e-15], (-z), If[LessEqual[y, 2.6e+96], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.30000000000000002e-15 or 2.6e96 < y

   1. Initial program 76.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

      2. lower-neg.f64 64.6

         \[\leadsto \color{blue}{-z} \]

   5. Applied rewrites 64.6%

      \[\leadsto \color{blue}{-z} \]

   if -1.30000000000000002e-15 < y < 2.6e96

   1. Initial program 98.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x + y} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{y + x} \]

      2. lower-+.f64 75.6

         \[\leadsto \color{blue}{y + x} \]

   5. Applied rewrites 75.6%

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

Alternative 8: 35.0% accurate, 9.7× speedup

Math

\[\begin{array}{l} \\ -z \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := (-z)

Derivation

1. Initial program 88.5%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(z\right)} \]

   2. lower-neg.f64 35.3

      \[\leadsto \color{blue}{-z} \]

5. Applied rewrites 35.3%

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

Developer Target 1: 94.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{-y} \cdot z\\ \mathbf{if}\;y < -3.7429310762689856 \cdot 10^{+171}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y < 3.5534662456086734 \cdot 10^{+168}:\\ \;\;\;\;\frac{x + y}{1 - \frac{y}{z}}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (/ (+ y x) (- y)) z)))
   (if (< y -3.7429310762689856e+171)
     t_0
     (if (< y 3.5534662456086734e+168) (/ (+ x y) (- 1.0 (/ y z))) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = ((y + x) / -y) * z
    if (y < (-3.7429310762689856d+171)) then
        tmp = t_0
    else if (y < 3.5534662456086734d+168) then
        tmp = (x + y) / (1.0d0 - (y / z))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = ((y + x) / -y) * z;
	double tmp;
	if (y < -3.7429310762689856e+171) {
		tmp = t_0;
	} else if (y < 3.5534662456086734e+168) {
		tmp = (x + y) / (1.0 - (y / z));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = ((y + x) / -y) * z
	tmp = 0
	if y < -3.7429310762689856e+171:
		tmp = t_0
	elif y < 3.5534662456086734e+168:
		tmp = (x + y) / (1.0 - (y / z))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(Float64(y + x) / Float64(-y)) * z)
	tmp = 0.0
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = Float64(Float64(x + y) / Float64(1.0 - Float64(y / z)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = ((y + x) / -y) * z;
	tmp = 0.0;
	if (y < -3.7429310762689856e+171)
		tmp = t_0;
	elseif (y < 3.5534662456086734e+168)
		tmp = (x + y) / (1.0 - (y / z));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(N[(y + x), $MachinePrecision] / (-y)), $MachinePrecision] * z), $MachinePrecision]}, If[Less[y, -3.7429310762689856e+171], t$95$0, If[Less[y, 3.5534662456086734e+168], N[(N[(x + y), $MachinePrecision] / N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Reproduce

herbie shell --seed 2024249 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Backend.Diagrams:calcFontMetrics from Chart-diagrams-1.5.1, A"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -3742931076268985600000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* (/ (+ y x) (- y)) z) (if (< y 3553466245608673400000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ (+ x y) (- 1 (/ y z))) (* (/ (+ y x) (- y)) z))))

  (/ (+ x y) (- 1.0 (/ y z))))