Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C

Percentage Accurate: 99.9% → 99.9%

Time: 8.3s

Alternatives: 9

Speedup: 1.1×

• 21.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

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 + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(y + z\right) + z \cdot 5 \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* x (+ y z)) (* z 5.0)))

C

double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

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 + z)) + (z * 5.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return (x * (y + z)) + (z * 5.0);
}

Python

def code(x, y, z):
	return (x * (y + z)) + (z * 5.0)

Julia

function code(x, y, z)
	return Float64(Float64(x * Float64(y + z)) + Float64(z * 5.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = (x * (y + z)) + (z * 5.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(x * N[(y + z), $MachinePrecision]), $MachinePrecision] + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y + z, x, z \cdot 5\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma (+ y z) x (* z 5.0)))

C

double code(double x, double y, double z) {
	return fma((y + z), x, (z * 5.0));
}

Julia

function code(x, y, z)
	return fma(Float64(y + z), x, Float64(z * 5.0))
end

Wolfram

code[x_, y_, z_] := N[(N[(y + z), $MachinePrecision] * x + N[(z * 5.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{\left(y + z\right) \cdot x} + z \cdot 5 \]

   2. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, x, z \cdot 5\right)} \]

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{y + z}, x, z \cdot 5\right) \]

   4. *-lowering-*.f64 100.0

      \[\leadsto \mathsf{fma}\left(y + z, x, \color{blue}{z \cdot 5}\right) \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, x, z \cdot 5\right)} \]
5. Add Preprocessing

Alternative 2: 53.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{-20}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq -6.5 \cdot 10^{-83}:\\ \;\;\;\;z \cdot 5\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-70}:\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -8.8e-20)
   (* y x)
   (if (<= y -6.5e-83) (* z 5.0) (if (<= y 1.15e-70) (* z x) (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-20) {
		tmp = y * x;
	} else if (y <= -6.5e-83) {
		tmp = z * 5.0;
	} else if (y <= 1.15e-70) {
		tmp = z * x;
	} else {
		tmp = 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) :: tmp
    if (y <= (-8.8d-20)) then
        tmp = y * x
    else if (y <= (-6.5d-83)) then
        tmp = z * 5.0d0
    else if (y <= 1.15d-70) then
        tmp = z * x
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -8.8e-20) {
		tmp = y * x;
	} else if (y <= -6.5e-83) {
		tmp = z * 5.0;
	} else if (y <= 1.15e-70) {
		tmp = z * x;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -8.8e-20:
		tmp = y * x
	elif y <= -6.5e-83:
		tmp = z * 5.0
	elif y <= 1.15e-70:
		tmp = z * x
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -8.8e-20)
		tmp = Float64(y * x);
	elseif (y <= -6.5e-83)
		tmp = Float64(z * 5.0);
	elseif (y <= 1.15e-70)
		tmp = Float64(z * x);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -8.8e-20)
		tmp = y * x;
	elseif (y <= -6.5e-83)
		tmp = z * 5.0;
	elseif (y <= 1.15e-70)
		tmp = z * x;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -8.8e-20], N[(y * x), $MachinePrecision], If[LessEqual[y, -6.5e-83], N[(z * 5.0), $MachinePrecision], If[LessEqual[y, 1.15e-70], N[(z * x), $MachinePrecision], N[(y * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.79999999999999964e-20 or 1.15e-70 < y

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 71.1

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 71.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 71.1

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

   7. Applied egg-rr 71.1%

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

   if -8.79999999999999964e-20 < y < -6.5e-83

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot z} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{5 \cdot z + 0} \]

      2. accelerator-lowering-fma.f64 75.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(5, z, 0\right)} \]

   5. Simplified 75.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{5 \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5} \]

      3. *-lowering-*.f64 75.4

         \[\leadsto \color{blue}{z \cdot 5} \]

   7. Applied egg-rr 75.4%

      \[\leadsto \color{blue}{z \cdot 5} \]

   if -6.5e-83 < y < 1.15e-70

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 69.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 69.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]

   6. Taylor expanded in z around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot z + 0} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot x} + 0 \]

      3. accelerator-lowering-fma.f64 55.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} \]

   8. Simplified 55.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x, 0\right)} \]
   9. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-lowering-*.f64 55.4

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

   10. Applied egg-rr 55.4%

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

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + z\right) \cdot x\\ \mathbf{if}\;x \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y z) x)))
   (if (<= x -5.0) t_0 (if (<= x 7.6e-20) (fma z 5.0 (* y x)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + z) * x;
	double tmp;
	if (x <= -5.0) {
		tmp = t_0;
	} else if (x <= 7.6e-20) {
		tmp = fma(z, 5.0, (y * x));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + z) * x)
	tmp = 0.0
	if (x <= -5.0)
		tmp = t_0;
	elseif (x <= 7.6e-20)
		tmp = fma(z, 5.0, Float64(y * x));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.0], t$95$0, If[LessEqual[x, 7.6e-20], N[(z * 5.0 + N[(y * x), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 7.5999999999999995e-20 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 98.9

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.9

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

   7. Applied egg-rr 98.9%

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

   if -5 < x < 7.5999999999999995e-20

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(z \cdot x + z \cdot 5\right) + y \cdot x} \]

      4. distribute-lft-out N/A

         \[\leadsto \color{blue}{z \cdot \left(x + 5\right)} + y \cdot x \]

      5. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

      6. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{x + 5}, y \cdot x\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{x \cdot y}\right) \]

      8. *-lowering-*.f64 100.0

         \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{x \cdot y}\right) \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, x \cdot y\right)} \]

   5. Taylor expanded in x around 0

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

   7. Simplified 99.7%

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{5}, x \cdot y\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -5:\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(z, 5, y \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + z\right) \cdot x\\ \mathbf{if}\;x \leq -5:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 7.6 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(y, x, z \cdot 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y z) x)))
   (if (<= x -5.0) t_0 (if (<= x 7.6e-20) (fma y x (* z 5.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + z) * x;
	double tmp;
	if (x <= -5.0) {
		tmp = t_0;
	} else if (x <= 7.6e-20) {
		tmp = fma(y, x, (z * 5.0));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + z) * x)
	tmp = 0.0
	if (x <= -5.0)
		tmp = t_0;
	elseif (x <= 7.6e-20)
		tmp = fma(y, x, Float64(z * 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -5.0], t$95$0, If[LessEqual[x, 7.6e-20], N[(y * x + N[(z * 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -5 or 7.5999999999999995e-20 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 98.9

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 98.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 98.9

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

   7. Applied egg-rr 98.9%

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

   if -5 < x < 7.5999999999999995e-20

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. *-commutative N/A

         \[\leadsto \color{blue}{\left(y + z\right) \cdot x} + z \cdot 5 \]

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, x, z \cdot 5\right)} \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{y + z}, x, z \cdot 5\right) \]

      4. *-lowering-*.f64 99.9

         \[\leadsto \mathsf{fma}\left(y + z, x, \color{blue}{z \cdot 5}\right) \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(y + z, x, z \cdot 5\right)} \]

   5. Taylor expanded in y around inf

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

   7. Simplified 99.7%

      \[\leadsto \mathsf{fma}\left(\color{blue}{y}, x, z \cdot 5\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 84.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(y + z\right) \cdot x\\ \mathbf{if}\;x \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-91}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ y z) x)))
   (if (<= x -3.2e-8) t_0 (if (<= x 2.2e-91) (* z (+ x 5.0)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (y + z) * x;
	double tmp;
	if (x <= -3.2e-8) {
		tmp = t_0;
	} else if (x <= 2.2e-91) {
		tmp = z * (x + 5.0);
	} 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 + z) * x
    if (x <= (-3.2d-8)) then
        tmp = t_0
    else if (x <= 2.2d-91) then
        tmp = z * (x + 5.0d0)
    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 + z) * x;
	double tmp;
	if (x <= -3.2e-8) {
		tmp = t_0;
	} else if (x <= 2.2e-91) {
		tmp = z * (x + 5.0);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (y + z) * x
	tmp = 0
	if x <= -3.2e-8:
		tmp = t_0
	elif x <= 2.2e-91:
		tmp = z * (x + 5.0)
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(y + z) * x)
	tmp = 0.0
	if (x <= -3.2e-8)
		tmp = t_0;
	elseif (x <= 2.2e-91)
		tmp = Float64(z * Float64(x + 5.0));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (y + z) * x;
	tmp = 0.0;
	if (x <= -3.2e-8)
		tmp = t_0;
	elseif (x <= 2.2e-91)
		tmp = z * (x + 5.0);
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(y + z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -3.2e-8], t$95$0, If[LessEqual[x, 2.2e-91], N[(z * N[(x + 5.0), $MachinePrecision]), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.2000000000000002e-8 or 2.2000000000000001e-91 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y + z, 0\right)} \]

      3. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

      4. +-lowering-+.f64 97.3

         \[\leadsto \mathsf{fma}\left(x, \color{blue}{z + y}, 0\right) \]

   5. Simplified 97.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, z + y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. +-commutative N/A

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

      5. +-lowering-+.f64 97.3

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

   7. Applied egg-rr 97.3%

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

   if -3.2000000000000002e-8 < x < 2.2000000000000001e-91

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto 5 \cdot z + \color{blue}{\left(x \cdot z + 0\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(5 \cdot z + x \cdot z\right) + 0} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} + 0 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5 + x, 0\right)} \]

      5. +-lowering-+.f64 69.8

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{5 + x}, 0\right) \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + 5\right)} \cdot z \]

      5. +-lowering-+.f64 69.8

         \[\leadsto \color{blue}{\left(x + 5\right)} \cdot z \]

   7. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{\left(x + 5\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.2 \cdot 10^{-8}:\\ \;\;\;\;\left(y + z\right) \cdot x\\ \mathbf{elif}\;x \leq 2.2 \cdot 10^{-91}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y + z\right) \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{-19}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+83}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= y -1.06e-19) (* y x) (if (<= y 3e+83) (* z (+ x 5.0)) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.06e-19) {
		tmp = y * x;
	} else if (y <= 3e+83) {
		tmp = z * (x + 5.0);
	} else {
		tmp = 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) :: tmp
    if (y <= (-1.06d-19)) then
        tmp = y * x
    else if (y <= 3d+83) then
        tmp = z * (x + 5.0d0)
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (y <= -1.06e-19) {
		tmp = y * x;
	} else if (y <= 3e+83) {
		tmp = z * (x + 5.0);
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if y <= -1.06e-19:
		tmp = y * x
	elif y <= 3e+83:
		tmp = z * (x + 5.0)
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (y <= -1.06e-19)
		tmp = Float64(y * x);
	elseif (y <= 3e+83)
		tmp = Float64(z * Float64(x + 5.0));
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (y <= -1.06e-19)
		tmp = y * x;
	elseif (y <= 3e+83)
		tmp = z * (x + 5.0);
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[y, -1.06e-19], N[(y * x), $MachinePrecision], If[LessEqual[y, 3e+83], N[(z * N[(x + 5.0), $MachinePrecision]), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.06e-19 or 3e83 < y

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 75.9

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 75.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 75.9

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

   7. Applied egg-rr 75.9%

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

   if -1.06e-19 < y < 3e83

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{5 \cdot z + x \cdot z} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto 5 \cdot z + \color{blue}{\left(x \cdot z + 0\right)} \]

      2. associate-+l+ N/A

         \[\leadsto \color{blue}{\left(5 \cdot z + x \cdot z\right) + 0} \]

      3. distribute-rgt-in N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} + 0 \]

      4. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5 + x, 0\right)} \]

      5. +-lowering-+.f64 80.3

         \[\leadsto \mathsf{fma}\left(z, \color{blue}{5 + x}, 0\right) \]

   5. Simplified 80.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, 5 + x, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{z \cdot \left(5 + x\right)} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \color{blue}{\left(5 + x\right) \cdot z} \]

      4. +-commutative N/A

         \[\leadsto \color{blue}{\left(x + 5\right)} \cdot z \]

      5. +-lowering-+.f64 80.3

         \[\leadsto \color{blue}{\left(x + 5\right)} \cdot z \]

   7. Applied egg-rr 80.3%

      \[\leadsto \color{blue}{\left(x + 5\right) \cdot z} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.06 \cdot 10^{-19}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+83}:\\ \;\;\;\;z \cdot \left(x + 5\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 60.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.9 \cdot 10^{-9}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 2.15 \cdot 10^{-85}:\\ \;\;\;\;z \cdot 5\\ \mathbf{else}:\\ \;\;\;\;y \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.9e-9) (* y x) (if (<= x 2.15e-85) (* z 5.0) (* y x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-9) {
		tmp = y * x;
	} else if (x <= 2.15e-85) {
		tmp = z * 5.0;
	} else {
		tmp = 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) :: tmp
    if (x <= (-1.9d-9)) then
        tmp = y * x
    else if (x <= 2.15d-85) then
        tmp = z * 5.0d0
    else
        tmp = y * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.9e-9) {
		tmp = y * x;
	} else if (x <= 2.15e-85) {
		tmp = z * 5.0;
	} else {
		tmp = y * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.9e-9:
		tmp = y * x
	elif x <= 2.15e-85:
		tmp = z * 5.0
	else:
		tmp = y * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.9e-9)
		tmp = Float64(y * x);
	elseif (x <= 2.15e-85)
		tmp = Float64(z * 5.0);
	else
		tmp = Float64(y * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.9e-9)
		tmp = y * x;
	elseif (x <= 2.15e-85)
		tmp = z * 5.0;
	else
		tmp = y * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.9e-9], N[(y * x), $MachinePrecision], If[LessEqual[x, 2.15e-85], N[(z * 5.0), $MachinePrecision], N[(y * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.90000000000000006e-9 or 2.14999999999999999e-85 < x

   1. Initial program 100.0%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{x \cdot y + 0} \]

      2. accelerator-lowering-fma.f64 55.4

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

   5. Simplified 55.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 55.4

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

   7. Applied egg-rr 55.4%

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

   if -1.90000000000000006e-9 < x < 2.14999999999999999e-85

   1. Initial program 99.9%

      \[x \cdot \left(y + z\right) + z \cdot 5 \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{5 \cdot z} \]
   4. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{5 \cdot z + 0} \]

      2. accelerator-lowering-fma.f64 69.8

         \[\leadsto \color{blue}{\mathsf{fma}\left(5, z, 0\right)} \]

   5. Simplified 69.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(5, z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

         \[\leadsto \color{blue}{5 \cdot z} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{z \cdot 5} \]

      3. *-lowering-*.f64 69.8

         \[\leadsto \color{blue}{z \cdot 5} \]

   7. Applied egg-rr 69.8%

      \[\leadsto \color{blue}{z \cdot 5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 98.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(z, x + 5, y \cdot x\right) \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (fma z (+ x 5.0) (* y x)))

C

double code(double x, double y, double z) {
	return fma(z, (x + 5.0), (y * x));
}

Julia

function code(x, y, z)
	return fma(z, Float64(x + 5.0), Float64(y * x))
end

Wolfram

code[x_, y_, z_] := N[(z * N[(x + 5.0), $MachinePrecision] + N[(y * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing
3. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left(y \cdot x + z \cdot x\right)} + z \cdot 5 \]

   2. associate-+l+ N/A

      \[\leadsto \color{blue}{y \cdot x + \left(z \cdot x + z \cdot 5\right)} \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(z \cdot x + z \cdot 5\right) + y \cdot x} \]

   4. distribute-lft-out N/A

      \[\leadsto \color{blue}{z \cdot \left(x + 5\right)} + y \cdot x \]

   5. accelerator-lowering-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, y \cdot x\right)} \]

   6. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{fma}\left(z, \color{blue}{x + 5}, y \cdot x\right) \]

   7. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{x \cdot y}\right) \]

   8. *-lowering-*.f64 99.6

      \[\leadsto \mathsf{fma}\left(z, x + 5, \color{blue}{x \cdot y}\right) \]

4. Applied egg-rr 99.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(z, x + 5, x \cdot y\right)} \]

5. Final simplification 99.6%

   \[\leadsto \mathsf{fma}\left(z, x + 5, y \cdot x\right) \]
6. Add Preprocessing

Alternative 9: 41.8% accurate, 2.8× speedup

Math

\[\begin{array}{l} \\ y \cdot x \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_] := N[(y * x), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[x \cdot \left(y + z\right) + z \cdot 5 \]
2. Add Preprocessing

3. Taylor expanded in y around inf

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

   1. +-rgt-identity N/A

      \[\leadsto \color{blue}{x \cdot y + 0} \]

   2. accelerator-lowering-fma.f64 46.8

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]

5. Simplified 46.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, y, 0\right)} \]
6. Step-by-step derivation

   1. +-rgt-identity N/A

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

   2. *-commutative N/A

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

   3. *-lowering-*.f64 46.8

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

7. Applied egg-rr 46.8%

   \[\leadsto \color{blue}{y \cdot x} \]
8. Add Preprocessing

Developer Target 1: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(x + 5\right) \cdot z + x \cdot y \end{array} \]

FPCore

(FPCore (x y z) :precision binary64 (+ (* (+ x 5.0) z) (* x y)))

C

double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

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 + 5.0d0) * z) + (x * y)
end function

Java

public static double code(double x, double y, double z) {
	return ((x + 5.0) * z) + (x * y);
}

Python

def code(x, y, z):
	return ((x + 5.0) * z) + (x * y)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x + 5.0) * z) + Float64(x * y))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x + 5.0) * z) + (x * y);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x + 5.0), $MachinePrecision] * z), $MachinePrecision] + N[(x * y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024195 
(FPCore (x y z)
  :name "Graphics.Rendering.Plot.Render.Plot.Legend:renderLegendOutside from plot-0.2.3.4, C"
  :precision binary64

  :alt
  (! :herbie-platform default (+ (* (+ x 5) z) (* x y)))

  (+ (* x (+ y z)) (* z 5.0)))