Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3

Percentage Accurate: 97.8% → 100.0%

Time: 6.9s

Alternatives: 10

Speedup: 1.4×

• 21.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot y + \left(x - 1\right) \cdot z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * 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) + ((x - 1.0d0) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot y + \left(x - 1\right) \cdot z \end{array} \]

FPCore

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

C

double code(double x, double y, double z) {
	return (x * y) + ((x - 1.0) * 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) + ((x - 1.0d0) * z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.4× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 98.0%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in z around 0

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

   1. sub-neg N/A

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

   2. metadata-eval N/A

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

   3. distribute-rgt-in N/A

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

   4. associate-+l+ N/A

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

   5. distribute-lft-in N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. +-commutative N/A

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

   9. lower-+.f64 N/A

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

   10. mul-1-neg N/A

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

   11. lower-neg.f64 100.0

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

5. Applied rewrites 100.0%

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

Alternative 2: 60.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;y \cdot x\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+114}:\\ \;\;\;\;y \cdot x\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.5e-15)
   (* y x)
   (if (<= x 1.2e-23) (- z) (if (<= x 6e+114) (* y x) (* z x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-15) {
		tmp = y * x;
	} else if (x <= 1.2e-23) {
		tmp = -z;
	} else if (x <= 6e+114) {
		tmp = y * x;
	} else {
		tmp = z * 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 <= (-1.5d-15)) then
        tmp = y * x
    else if (x <= 1.2d-23) then
        tmp = -z
    else if (x <= 6d+114) then
        tmp = y * x
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.5e-15) {
		tmp = y * x;
	} else if (x <= 1.2e-23) {
		tmp = -z;
	} else if (x <= 6e+114) {
		tmp = y * x;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.5e-15:
		tmp = y * x
	elif x <= 1.2e-23:
		tmp = -z
	elif x <= 6e+114:
		tmp = y * x
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.5e-15)
		tmp = Float64(y * x);
	elseif (x <= 1.2e-23)
		tmp = Float64(-z);
	elseif (x <= 6e+114)
		tmp = Float64(y * x);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.5e-15)
		tmp = y * x;
	elseif (x <= 1.2e-23)
		tmp = -z;
	elseif (x <= 6e+114)
		tmp = y * x;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.5e-15], N[(y * x), $MachinePrecision], If[LessEqual[x, 1.2e-23], (-z), If[LessEqual[x, 6e+114], N[(y * x), $MachinePrecision], N[(z * x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.5e-15 or 1.19999999999999998e-23 < x < 6.0000000000000001e114

   1. Initial program 96.7%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 64.2

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

   5. Applied rewrites 64.2%

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

   if -1.5e-15 < x < 1.19999999999999998e-23

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.1

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

   5. Applied rewrites 70.1%

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

   if 6.0000000000000001e114 < x

   1. Initial program 95.5%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 69.5%

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

Alternative 3: 98.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;\left(z + y\right) \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, y \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.0)
   (* (+ z y) x)
   (if (<= x 1.0) (fma y x (- z)) (fma z x (* y x)))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.0) {
		tmp = (z + y) * x;
	} else if (x <= 1.0) {
		tmp = fma(y, x, -z);
	} else {
		tmp = fma(z, x, (y * x));
	}
	return tmp;
}

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(Float64(z + y) * x);
	elseif (x <= 1.0)
		tmp = fma(y, x, Float64(-z));
	else
		tmp = fma(z, x, Float64(y * x));
	end
	return tmp
end

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if x < -1

   1. Initial program 96.6%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 100.0

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

   5. Applied rewrites 100.0%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   7. Applied rewrites 99.3%

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

   if 1 < x

   1. Initial program 95.6%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.5

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

   5. Applied rewrites 99.5%

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

   7. Applied rewrites 99.5%

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

Alternative 4: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + y\right) \cdot x\\ \mathbf{if}\;x \leq -1:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;\mathsf{fma}\left(y, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z y) x)))
   (if (<= x -1.0) t_0 (if (<= x 1.0) (fma y x (- z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z + y) * x;
	double tmp;
	if (x <= -1.0) {
		tmp = t_0;
	} else if (x <= 1.0) {
		tmp = fma(y, x, -z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 96.1%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.3

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

   7. Applied rewrites 99.3%

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

Alternative 5: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + y\right) \cdot x\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(z, x, -z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z y) x)))
   (if (<= y -3.7e-40) t_0 (if (<= y 1.55e+25) (fma z x (- z)) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z + y) * x;
	double tmp;
	if (y <= -3.7e-40) {
		tmp = t_0;
	} else if (y <= 1.55e+25) {
		tmp = fma(z, x, -z);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + y) * x)
	tmp = 0.0
	if (y <= -3.7e-40)
		tmp = t_0;
	elseif (y <= 1.55e+25)
		tmp = fma(z, x, Float64(-z));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.7e-40], t$95$0, If[LessEqual[y, 1.55e+25], N[(z * x + (-z)), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.69999999999999998e-40 or 1.5499999999999999e25 < y

   1. Initial program 96.1%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -3.69999999999999998e-40 < y < 1.5499999999999999e25

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 100.0

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 86.6

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

   7. Applied rewrites 86.6%

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

   9. Applied rewrites 86.7%

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

Alternative 6: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + y\right) \cdot x\\ \mathbf{if}\;y \leq -3.7 \cdot 10^{-40}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 1.55 \cdot 10^{+25}:\\ \;\;\;\;\left(x - 1\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z y) x)))
   (if (<= y -3.7e-40) t_0 (if (<= y 1.55e+25) (* (- x 1.0) z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z + y) * x;
	double tmp;
	if (y <= -3.7e-40) {
		tmp = t_0;
	} else if (y <= 1.55e+25) {
		tmp = (x - 1.0) * 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 = (z + y) * x
    if (y <= (-3.7d-40)) then
        tmp = t_0
    else if (y <= 1.55d+25) then
        tmp = (x - 1.0d0) * 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 = (z + y) * x;
	double tmp;
	if (y <= -3.7e-40) {
		tmp = t_0;
	} else if (y <= 1.55e+25) {
		tmp = (x - 1.0) * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + y) * x
	tmp = 0
	if y <= -3.7e-40:
		tmp = t_0
	elif y <= 1.55e+25:
		tmp = (x - 1.0) * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + y) * x)
	tmp = 0.0
	if (y <= -3.7e-40)
		tmp = t_0;
	elseif (y <= 1.55e+25)
		tmp = Float64(Float64(x - 1.0) * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + y) * x;
	tmp = 0.0;
	if (y <= -3.7e-40)
		tmp = t_0;
	elseif (y <= 1.55e+25)
		tmp = (x - 1.0) * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -3.7e-40], t$95$0, If[LessEqual[y, 1.55e+25], N[(N[(x - 1.0), $MachinePrecision] * z), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.69999999999999998e-40 or 1.5499999999999999e25 < y

   1. Initial program 96.1%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 85.5

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

   5. Applied rewrites 85.5%

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

   if -3.69999999999999998e-40 < y < 1.5499999999999999e25

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 86.6

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

   5. Applied rewrites 86.6%

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

Alternative 7: 84.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \left(z + y\right) \cdot x\\ \mathbf{if}\;x \leq -1.5 \cdot 10^{-15}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 1.2 \cdot 10^{-23}:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (let* ((t_0 (* (+ z y) x)))
   (if (<= x -1.5e-15) t_0 (if (<= x 1.2e-23) (- z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = (z + y) * x;
	double tmp;
	if (x <= -1.5e-15) {
		tmp = t_0;
	} else if (x <= 1.2e-23) {
		tmp = -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 = (z + y) * x
    if (x <= (-1.5d-15)) then
        tmp = t_0
    else if (x <= 1.2d-23) then
        tmp = -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 = (z + y) * x;
	double tmp;
	if (x <= -1.5e-15) {
		tmp = t_0;
	} else if (x <= 1.2e-23) {
		tmp = -z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = (z + y) * x
	tmp = 0
	if x <= -1.5e-15:
		tmp = t_0
	elif x <= 1.2e-23:
		tmp = -z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(Float64(z + y) * x)
	tmp = 0.0
	if (x <= -1.5e-15)
		tmp = t_0;
	elseif (x <= 1.2e-23)
		tmp = Float64(-z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = (z + y) * x;
	tmp = 0.0;
	if (x <= -1.5e-15)
		tmp = t_0;
	elseif (x <= 1.2e-23)
		tmp = -z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(N[(z + y), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -1.5e-15], t$95$0, If[LessEqual[x, 1.2e-23], (-z), t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.5e-15 or 1.19999999999999998e-23 < x

   1. Initial program 96.3%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 98.1

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

   5. Applied rewrites 98.1%

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

   if -1.5e-15 < x < 1.19999999999999998e-23

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 70.1

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

   5. Applied rewrites 70.1%

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

Alternative 8: 60.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;x \leq 1:\\ \;\;\;\;-z\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z)
 :precision binary64
 (if (<= x -1.0) (* z x) (if (<= x 1.0) (- z) (* z x))))

C

double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.0) {
		tmp = z * x;
	} else if (x <= 1.0) {
		tmp = -z;
	} else {
		tmp = z * 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 <= (-1.0d0)) then
        tmp = z * x
    else if (x <= 1.0d0) then
        tmp = -z
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double tmp;
	if (x <= -1.0) {
		tmp = z * x;
	} else if (x <= 1.0) {
		tmp = -z;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z):
	tmp = 0
	if x <= -1.0:
		tmp = z * x
	elif x <= 1.0:
		tmp = -z
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z)
	tmp = 0.0
	if (x <= -1.0)
		tmp = Float64(z * x);
	elseif (x <= 1.0)
		tmp = Float64(-z);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	tmp = 0.0;
	if (x <= -1.0)
		tmp = z * x;
	elseif (x <= 1.0)
		tmp = -z;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := If[LessEqual[x, -1.0], N[(z * x), $MachinePrecision], If[LessEqual[x, 1.0], (-z), N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1 or 1 < x

   1. Initial program 96.1%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. +-commutative N/A

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

      4. lower-+.f64 99.7

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

   5. Applied rewrites 99.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 53.4%

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

   if -1 < x < 1

   1. Initial program 100.0%

      \[x \cdot y + \left(x - 1\right) \cdot z \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 66.4

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

   5. Applied rewrites 66.4%

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

Alternative 9: 36.2% accurate, 5.7× 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 98.0%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 34.6

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

5. Applied rewrites 34.6%

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

Alternative 10: 2.6% accurate, 17.0× 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 z
end

MATLAB

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

Wolfram

code[x_, y_, z_] := z

Derivation

1. Initial program 98.0%

   \[x \cdot y + \left(x - 1\right) \cdot z \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. lower-neg.f64 34.6

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

5. Applied rewrites 34.6%

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

7. Applied rewrites 2.6%

   \[\leadsto z \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z)
  :name "Graphics.Rendering.Chart.Drawing:drawTextsR from Chart-1.5.3"
  :precision binary64
  (+ (* x y) (* (- x 1.0) z)))