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

Percentage Accurate: 88.7% → 99.5%

Time: 6.6s

Alternatives: 7

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: 88.7% 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: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-257}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-x}{y} \cdot z - z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (/ (+ y x) (- 1.0 (/ y z)))))
   (if (<= t_0 -2e-257)
     t_0
     (if (<= t_0 0.0) (- (* (/ (- x) y) z) z) (* (/ z (- z y)) (+ y x))))))

C

double code(double x, double y, double z) {
	double t_0 = (y + x) / (1.0 - (y / z));
	double tmp;
	if (t_0 <= -2e-257) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((-x / y) * z) - z;
	} else {
		tmp = (z / (z - y)) * (y + x);
	}
	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 <= (-2d-257)) then
        tmp = t_0
    else if (t_0 <= 0.0d0) then
        tmp = ((-x / y) * z) - z
    else
        tmp = (z / (z - y)) * (y + x)
    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 <= -2e-257) {
		tmp = t_0;
	} else if (t_0 <= 0.0) {
		tmp = ((-x / y) * z) - z;
	} else {
		tmp = (z / (z - y)) * (y + x);
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + x) / (1.0 - (y / z))
	tmp = 0
	if t_0 <= -2e-257:
		tmp = t_0
	elif t_0 <= 0.0:
		tmp = ((-x / y) * z) - z
	else:
		tmp = (z / (z - y)) * (y + x)
	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 <= -2e-257)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(Float64(Float64(-x) / y) * z) - z);
	else
		tmp = Float64(Float64(z / Float64(z - y)) * Float64(y + x));
	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 <= -2e-257)
		tmp = t_0;
	elseif (t_0 <= 0.0)
		tmp = ((-x / y) * z) - z;
	else
		tmp = (z / (z - y)) * (y + x);
	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, -2e-257], t$95$0, If[LessEqual[t$95$0, 0.0], N[(N[(N[((-x) / y), $MachinePrecision] * z), $MachinePrecision] - z), $MachinePrecision], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(y + x), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < -2e-257

   1. Initial program 99.9%

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

   if -2e-257 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 8.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   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. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-inverses N/A

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

      7. lift-/.f64 N/A

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

      8. div-sub N/A

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

      9. lift--.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{y + x}{1 - \frac{y}{z}}\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{-x}{y} \cdot z - z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ t_1 := \frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-257}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\frac{-x}{y} \cdot z - z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e-257) 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 <= -2e-257) {
		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 <= (-2d-257)) 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 <= -2e-257) {
		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 <= -2e-257:
		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 <= -2e-257)
		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 <= -2e-257)
		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, -2e-257], 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))) < -2e-257 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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-inverses N/A

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

      7. lift-/.f64 N/A

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

      8. div-sub N/A

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

      9. lift--.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -2e-257 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 8.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\frac{-x}{y} \cdot z - z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{y + x}{1 - \frac{y}{z}}\\ t_1 := \frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{if}\;t\_0 \leq -2 \cdot 10^{-257}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 0:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e-257) t_1 (if (<= t_0 0.0) (* (- -1.0 (/ x y)) 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 <= -2e-257) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 - (x / y)) * 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 <= (-2d-257)) then
        tmp = t_1
    else if (t_0 <= 0.0d0) then
        tmp = ((-1.0d0) - (x / y)) * 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 <= -2e-257) {
		tmp = t_1;
	} else if (t_0 <= 0.0) {
		tmp = (-1.0 - (x / y)) * 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 <= -2e-257:
		tmp = t_1
	elif t_0 <= 0.0:
		tmp = (-1.0 - (x / y)) * 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 <= -2e-257)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = Float64(Float64(-1.0 - Float64(x / y)) * 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 <= -2e-257)
		tmp = t_1;
	elseif (t_0 <= 0.0)
		tmp = (-1.0 - (x / y)) * 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, -2e-257], t$95$1, If[LessEqual[t$95$0, 0.0], N[(N[(-1.0 - N[(x / y), $MachinePrecision]), $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))) < -2e-257 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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-inverses N/A

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

      7. lift-/.f64 N/A

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

      8. div-sub N/A

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

      9. lift--.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 99.8

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.8

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

   6. Applied rewrites 99.8%

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

   if -2e-257 < (/.f64 (+.f64 x y) (-.f64 #s(literal 1 binary64) (/.f64 y z))) < 0.0

   1. Initial program 8.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 99.9

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

   5. Applied rewrites 99.9%

      \[\leadsto \color{blue}{\left(-1 - \frac{x}{y}\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y + x}{1 - \frac{y}{z}} \leq -2 \cdot 10^{-257}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \mathbf{elif}\;\frac{y + x}{1 - \frac{y}{z}} \leq 0:\\ \;\;\;\;\left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - y} \cdot \left(y + x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 73.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(-1 - \frac{x}{y}\right) \cdot z\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+42}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{z}{z - y} \cdot x\\ \mathbf{elif}\;y \leq 1.75 \cdot 10^{+55}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (- -1.0 (/ x y)) z)))
   (if (<= y -1.1e+42)
     t_0
     (if (<= y -5.8e-30)
       (+ y x)
       (if (<= y 9.6e-42)
         (* (/ z (- z y)) x)
         (if (<= y 1.75e+55) (+ y x) t_0))))))

C

double code(double x, double y, double z) {
	double t_0 = (-1.0 - (x / y)) * z;
	double tmp;
	if (y <= -1.1e+42) {
		tmp = t_0;
	} else if (y <= -5.8e-30) {
		tmp = y + x;
	} else if (y <= 9.6e-42) {
		tmp = (z / (z - y)) * x;
	} else if (y <= 1.75e+55) {
		tmp = y + x;
	} 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 <= (-1.1d+42)) then
        tmp = t_0
    else if (y <= (-5.8d-30)) then
        tmp = y + x
    else if (y <= 9.6d-42) then
        tmp = (z / (z - y)) * x
    else if (y <= 1.75d+55) then
        tmp = y + x
    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 <= -1.1e+42) {
		tmp = t_0;
	} else if (y <= -5.8e-30) {
		tmp = y + x;
	} else if (y <= 9.6e-42) {
		tmp = (z / (z - y)) * x;
	} else if (y <= 1.75e+55) {
		tmp = y + x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (-1.0 - (x / y)) * z
	tmp = 0
	if y <= -1.1e+42:
		tmp = t_0
	elif y <= -5.8e-30:
		tmp = y + x
	elif y <= 9.6e-42:
		tmp = (z / (z - y)) * x
	elif y <= 1.75e+55:
		tmp = y + x
	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 <= -1.1e+42)
		tmp = t_0;
	elseif (y <= -5.8e-30)
		tmp = Float64(y + x);
	elseif (y <= 9.6e-42)
		tmp = Float64(Float64(z / Float64(z - y)) * x);
	elseif (y <= 1.75e+55)
		tmp = Float64(y + x);
	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 <= -1.1e+42)
		tmp = t_0;
	elseif (y <= -5.8e-30)
		tmp = y + x;
	elseif (y <= 9.6e-42)
		tmp = (z / (z - y)) * x;
	elseif (y <= 1.75e+55)
		tmp = y + x;
	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, -1.1e+42], t$95$0, If[LessEqual[y, -5.8e-30], N[(y + x), $MachinePrecision], If[LessEqual[y, 9.6e-42], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.75e+55], N[(y + x), $MachinePrecision], t$95$0]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.1000000000000001e42 or 1.75000000000000005e55 < y

   1. Initial program 68.5%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 83.0

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

   5. Applied rewrites 83.0%

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

   if -1.1000000000000001e42 < y < -5.79999999999999978e-30 or 9.60000000000000011e-42 < y < 1.75000000000000005e55

   1. Initial program 100.0%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 81.2

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

   5. Applied rewrites 81.2%

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

   if -5.79999999999999978e-30 < y < 9.60000000000000011e-42

   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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-inverses N/A

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

      7. lift-/.f64 N/A

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

      8. div-sub N/A

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

      9. lift--.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.2

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

   9. Applied rewrites 84.2%

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

Alternative 5: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+46}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq -5.8 \cdot 10^{-30}:\\ \;\;\;\;y + x\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{-42}:\\ \;\;\;\;\frac{z}{z - y} \cdot x\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+115}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+46)
   (- z)
   (if (<= y -5.8e-30)
     (+ y x)
     (if (<= y 9.6e-42)
       (* (/ z (- z y)) x)
       (if (<= y 1.8e+115) (+ y x) (- z))))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+46) {
		tmp = -z;
	} else if (y <= -5.8e-30) {
		tmp = y + x;
	} else if (y <= 9.6e-42) {
		tmp = (z / (z - y)) * x;
	} else if (y <= 1.8e+115) {
		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 <= (-1.7d+46)) then
        tmp = -z
    else if (y <= (-5.8d-30)) then
        tmp = y + x
    else if (y <= 9.6d-42) then
        tmp = (z / (z - y)) * x
    else if (y <= 1.8d+115) 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 <= -1.7e+46) {
		tmp = -z;
	} else if (y <= -5.8e-30) {
		tmp = y + x;
	} else if (y <= 9.6e-42) {
		tmp = (z / (z - y)) * x;
	} else if (y <= 1.8e+115) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.7e+46:
		tmp = -z
	elif y <= -5.8e-30:
		tmp = y + x
	elif y <= 9.6e-42:
		tmp = (z / (z - y)) * x
	elif y <= 1.8e+115:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+46)
		tmp = Float64(-z);
	elseif (y <= -5.8e-30)
		tmp = Float64(y + x);
	elseif (y <= 9.6e-42)
		tmp = Float64(Float64(z / Float64(z - y)) * x);
	elseif (y <= 1.8e+115)
		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 <= -1.7e+46)
		tmp = -z;
	elseif (y <= -5.8e-30)
		tmp = y + x;
	elseif (y <= 9.6e-42)
		tmp = (z / (z - y)) * x;
	elseif (y <= 1.8e+115)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e+46], (-z), If[LessEqual[y, -5.8e-30], N[(y + x), $MachinePrecision], If[LessEqual[y, 9.6e-42], N[(N[(z / N[(z - y), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.8e+115], N[(y + x), $MachinePrecision], (-z)]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.6999999999999999e46 or 1.8e115 < y

   1. Initial program 65.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 73.3

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

   5. Applied rewrites 73.3%

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

   if -1.6999999999999999e46 < y < -5.79999999999999978e-30 or 9.60000000000000011e-42 < y < 1.8e115

   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 73.8

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

   5. Applied rewrites 73.8%

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

   if -5.79999999999999978e-30 < y < 9.60000000000000011e-42

   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

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 99.6

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

      5. lift-+.f64 N/A

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

      6. +-commutative N/A

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

      7. lower-+.f64 99.6

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

   4. Applied rewrites 99.6%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lower-*.f64 N/A

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

      5. lift--.f64 N/A

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

      6. *-inverses N/A

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

      7. lift-/.f64 N/A

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

      8. div-sub N/A

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

      9. lift--.f64 N/A

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

      10. clear-num N/A

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

      11. lower-/.f64 99.9

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. lower-+.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in x around inf

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

      1. *-commutative N/A

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

      2. associate-*l/ N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 84.2

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

   9. Applied rewrites 84.2%

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

Alternative 6: 67.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.7 \cdot 10^{+46}:\\ \;\;\;\;-z\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{+115}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;-z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.7e+46) (- z) (if (<= y 1.8e+115) (+ y x) (- z))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.7e+46) {
		tmp = -z;
	} else if (y <= 1.8e+115) {
		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 <= (-1.7d+46)) then
        tmp = -z
    else if (y <= 1.8d+115) 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 <= -1.7e+46) {
		tmp = -z;
	} else if (y <= 1.8e+115) {
		tmp = y + x;
	} else {
		tmp = -z;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.7e+46:
		tmp = -z
	elif y <= 1.8e+115:
		tmp = y + x
	else:
		tmp = -z
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.7e+46)
		tmp = Float64(-z);
	elseif (y <= 1.8e+115)
		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 <= -1.7e+46)
		tmp = -z;
	elseif (y <= 1.8e+115)
		tmp = y + x;
	else
		tmp = -z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.7e+46], (-z), If[LessEqual[y, 1.8e+115], N[(y + x), $MachinePrecision], (-z)]]

Derivation

1. Split input into 2 regimes

2. if y < -1.6999999999999999e46 or 1.8e115 < y

   1. Initial program 65.3%

      \[\frac{x + y}{1 - \frac{y}{z}} \]
   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 73.3

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

   5. Applied rewrites 73.3%

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

   if -1.6999999999999999e46 < y < 1.8e115

   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 71.3

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

   5. Applied rewrites 71.3%

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

Alternative 7: 35.2% 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 86.7%

   \[\frac{x + y}{1 - \frac{y}{z}} \]
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 35.2

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

5. Applied rewrites 35.2%

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

Developer Target 1: 93.8% 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 2024276 
(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))))