(/ (* PI (+ (* -1/2 z0) 1/4)) z1)

Percentage Accurate: 99.6% → 99.6%

Time: 1.7s

Alternatives: 5

Speedup: N/A×

Specification

Math

\[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (/ (* PI (+ (* -0.5 z0) 0.25)) z1))

C

double code(double z0, double z1) {
	return (((double) M_PI) * ((-0.5 * z0) + 0.25)) / z1;
}

Java

public static double code(double z0, double z1) {
	return (Math.PI * ((-0.5 * z0) + 0.25)) / z1;
}

Python

def code(z0, z1):
	return (math.pi * ((-0.5 * z0) + 0.25)) / z1

Julia

function code(z0, z1)
	return Float64(Float64(pi * Float64(Float64(-0.5 * z0) + 0.25)) / z1)
end

MATLAB

function tmp = code(z0, z1)
	tmp = (pi * ((-0.5 * z0) + 0.25)) / z1;
end

Wolfram

code[z0_, z1_] := N[(N[(Pi * N[(N[(-0.5 * z0), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Initial Program: 99.6% accurate, 1.0× speedup

Math

\[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (/ (* PI (+ (* -0.5 z0) 0.25)) z1))

C

double code(double z0, double z1) {
	return (((double) M_PI) * ((-0.5 * z0) + 0.25)) / z1;
}

Java

public static double code(double z0, double z1) {
	return (Math.PI * ((-0.5 * z0) + 0.25)) / z1;
}

Python

def code(z0, z1):
	return (math.pi * ((-0.5 * z0) + 0.25)) / z1

Julia

function code(z0, z1)
	return Float64(Float64(pi * Float64(Float64(-0.5 * z0) + 0.25)) / z1)
end

MATLAB

function tmp = code(z0, z1)
	tmp = (pi * ((-0.5 * z0) + 0.25)) / z1;
end

Wolfram

code[z0_, z1_] := N[(N[(Pi * N[(N[(-0.5 * z0), $MachinePrecision] + 0.25), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]

Alternative 1: 99.6% accurate, 0.8× speedup

Math

\[\frac{\pi}{\frac{z1}{-0.5 \cdot z0 - -0.25}} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (/ PI (/ z1 (- (* -0.5 z0) -0.25))))

C

double code(double z0, double z1) {
	return ((double) M_PI) / (z1 / ((-0.5 * z0) - -0.25));
}

Java

public static double code(double z0, double z1) {
	return Math.PI / (z1 / ((-0.5 * z0) - -0.25));
}

Python

def code(z0, z1):
	return math.pi / (z1 / ((-0.5 * z0) - -0.25))

Julia

function code(z0, z1)
	return Float64(pi / Float64(z1 / Float64(Float64(-0.5 * z0) - -0.25)))
end

MATLAB

function tmp = code(z0, z1)
	tmp = pi / (z1 / ((-0.5 * z0) - -0.25));
end

Wolfram

code[z0_, z1_] := N[(Pi / N[(z1 / N[(N[(-0.5 * z0), $MachinePrecision] - -0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\pi \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) + \pi \cdot \frac{1}{4}}}{z1} \]

   4. add-flip N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   11. *-commutative N/A

       \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \pi}\right)\right)}{z1} \]

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

       \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

   14. metadata-eval 99.6%

       \[\leadsto \frac{\left(z0 \cdot -0.5\right) \cdot \pi - \color{blue}{-0.25} \cdot \pi}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\left(z0 \cdot -0.5\right) \cdot \pi - -0.25 \cdot \pi}}{z1} \]
4. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \frac{-1}{4} \cdot \pi}{z1}} \]

   2. div-flip-rev N/A

      \[\leadsto \color{blue}{\frac{1}{\frac{z1}{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \frac{-1}{4} \cdot \pi}}} \]

   3. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \frac{-1}{4} \cdot \pi}}} \]

   4. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi} - \frac{-1}{4} \cdot \pi}} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\frac{-1}{4} \cdot \pi}}} \]

   6. distribute-rgt-out-- N/A

      \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\pi \cdot \left(z0 \cdot \frac{-1}{2} - \frac{-1}{4}\right)}}} \]

   7. lift--.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \color{blue}{\left(z0 \cdot \frac{-1}{2} - \frac{-1}{4}\right)}}} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \left(\color{blue}{z0 \cdot \frac{-1}{2}} - \frac{-1}{4}\right)}} \]

   9. *-commutative N/A

      \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \left(\color{blue}{\frac{-1}{2} \cdot z0} - \frac{-1}{4}\right)}} \]

   10. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \left(\color{blue}{\frac{-1}{2} \cdot z0} - \frac{-1}{4}\right)}} \]

   11. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\left(\frac{-1}{2} \cdot z0 - \frac{-1}{4}\right) \cdot \pi}}} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\left(\frac{-1}{2} \cdot z0 - \frac{-1}{4}\right) \cdot \pi}}} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\left(\frac{-1}{2} \cdot z0 - \frac{-1}{4}\right) \cdot \pi}}} \]

   14. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0 - \frac{-1}{4}\right)}}} \]

   15. lift--.f64 N/A

       \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 - \frac{-1}{4}\right)}}} \]

   16. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \left(\color{blue}{\frac{-1}{2} \cdot z0} - \frac{-1}{4}\right)}} \]

   17. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \left(\color{blue}{z0 \cdot \frac{-1}{2}} - \frac{-1}{4}\right)}} \]

   18. lift-*.f64 N/A

       \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \left(\color{blue}{z0 \cdot \frac{-1}{2}} - \frac{-1}{4}\right)}} \]

   19. lift--.f64 N/A

       \[\leadsto \frac{1}{\frac{z1}{\pi \cdot \color{blue}{\left(z0 \cdot \frac{-1}{2} - \frac{-1}{4}\right)}}} \]

   20. *-commutative N/A

       \[\leadsto \frac{1}{\frac{z1}{\color{blue}{\left(z0 \cdot \frac{-1}{2} - \frac{-1}{4}\right) \cdot \pi}}} \]

5. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\pi}{\frac{z1}{-0.5 \cdot z0 - -0.25}}} \]
6. Add Preprocessing

Alternative 2: 99.6% accurate, 1.0× speedup

Math

\[\frac{\pi}{z1} \cdot \left(z0 \cdot -0.5 - -0.25\right) \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (* (/ PI z1) (- (* z0 -0.5) -0.25)))

C

double code(double z0, double z1) {
	return (((double) M_PI) / z1) * ((z0 * -0.5) - -0.25);
}

Java

public static double code(double z0, double z1) {
	return (Math.PI / z1) * ((z0 * -0.5) - -0.25);
}

Python

def code(z0, z1):
	return (math.pi / z1) * ((z0 * -0.5) - -0.25)

Julia

function code(z0, z1)
	return Float64(Float64(pi / z1) * Float64(Float64(z0 * -0.5) - -0.25))
end

MATLAB

function tmp = code(z0, z1)
	tmp = (pi / z1) * ((z0 * -0.5) - -0.25);
end

Wolfram

code[z0_, z1_] := N[(N[(Pi / z1), $MachinePrecision] * N[(N[(z0 * -0.5), $MachinePrecision] - -0.25), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]
2. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\pi \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}{z1}} \]

   2. mult-flip N/A

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)\right) \cdot \frac{1}{z1}} \]

   3. lift-*.f64 N/A

      \[\leadsto \color{blue}{\left(\pi \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)\right)} \cdot \frac{1}{z1} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right) \cdot \pi\right)} \cdot \frac{1}{z1} \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right) \cdot \left(\pi \cdot \frac{1}{z1}\right)} \]

   6. *-commutative N/A

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{1}{z1}\right) \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)} \]

   7. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\pi \cdot \frac{1}{z1}\right) \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)} \]

   8. mult-flip-rev N/A

      \[\leadsto \color{blue}{\frac{\pi}{z1}} \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right) \]

   9. lower-/.f64 99.6%

      \[\leadsto \color{blue}{\frac{\pi}{z1}} \cdot \left(-0.5 \cdot z0 + 0.25\right) \]

   10. lift-+.f64 N/A

       \[\leadsto \frac{\pi}{z1} \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)} \]

   11. add-flip N/A

       \[\leadsto \frac{\pi}{z1} \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 - \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \]

   12. lower--.f64 N/A

       \[\leadsto \frac{\pi}{z1} \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 - \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right)} \]

   13. lift-*.f64 N/A

       \[\leadsto \frac{\pi}{z1} \cdot \left(\color{blue}{\frac{-1}{2} \cdot z0} - \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \]

   14. *-commutative N/A

       \[\leadsto \frac{\pi}{z1} \cdot \left(\color{blue}{z0 \cdot \frac{-1}{2}} - \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \]

   15. lower-*.f64 N/A

       \[\leadsto \frac{\pi}{z1} \cdot \left(\color{blue}{z0 \cdot \frac{-1}{2}} - \left(\mathsf{neg}\left(\frac{1}{4}\right)\right)\right) \]

   16. metadata-eval 99.6%

       \[\leadsto \frac{\pi}{z1} \cdot \left(z0 \cdot -0.5 - \color{blue}{-0.25}\right) \]

3. Applied rewrites 99.6%

   \[\leadsto \color{blue}{\frac{\pi}{z1} \cdot \left(z0 \cdot -0.5 - -0.25\right)} \]
4. Add Preprocessing

Alternative 3: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z0 \leq -1100:\\ \;\;\;\;z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1}\right)\\ \mathbf{elif}\;z0 \leq 13.5:\\ \;\;\;\;\frac{0.7853981633974483}{z1}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.5 \cdot \left(z0 \cdot \pi\right)}{z1}\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (if (<= z0 -1100.0)
  (* z0 (* -0.5 (/ PI z1)))
  (if (<= z0 13.5)
    (/ 0.7853981633974483 z1)
    (/ (* -0.5 (* z0 PI)) z1))))

C

double code(double z0, double z1) {
	double tmp;
	if (z0 <= -1100.0) {
		tmp = z0 * (-0.5 * (((double) M_PI) / z1));
	} else if (z0 <= 13.5) {
		tmp = 0.7853981633974483 / z1;
	} else {
		tmp = (-0.5 * (z0 * ((double) M_PI))) / z1;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double tmp;
	if (z0 <= -1100.0) {
		tmp = z0 * (-0.5 * (Math.PI / z1));
	} else if (z0 <= 13.5) {
		tmp = 0.7853981633974483 / z1;
	} else {
		tmp = (-0.5 * (z0 * Math.PI)) / z1;
	}
	return tmp;
}

Python

def code(z0, z1):
	tmp = 0
	if z0 <= -1100.0:
		tmp = z0 * (-0.5 * (math.pi / z1))
	elif z0 <= 13.5:
		tmp = 0.7853981633974483 / z1
	else:
		tmp = (-0.5 * (z0 * math.pi)) / z1
	return tmp

Julia

function code(z0, z1)
	tmp = 0.0
	if (z0 <= -1100.0)
		tmp = Float64(z0 * Float64(-0.5 * Float64(pi / z1)));
	elseif (z0 <= 13.5)
		tmp = Float64(0.7853981633974483 / z1);
	else
		tmp = Float64(Float64(-0.5 * Float64(z0 * pi)) / z1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	tmp = 0.0;
	if (z0 <= -1100.0)
		tmp = z0 * (-0.5 * (pi / z1));
	elseif (z0 <= 13.5)
		tmp = 0.7853981633974483 / z1;
	else
		tmp = (-0.5 * (z0 * pi)) / z1;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := If[LessEqual[z0, -1100.0], N[(z0 * N[(-0.5 * N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z0, 13.5], N[(0.7853981633974483 / z1), $MachinePrecision], N[(N[(-0.5 * N[(z0 * Pi), $MachinePrecision]), $MachinePrecision] / z1), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z0 < -1100

   1. Initial program 99.6%

      \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto \color{blue}{z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\pi}{z0 \cdot z1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \color{blue}{\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \color{blue}{\frac{1}{4}} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z0 \cdot z1}}\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\pi}{\color{blue}{z0} \cdot z1}\right) \]

      9. lower-*.f64 87.8%

         \[\leadsto z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1} + 0.25 \cdot \frac{\pi}{z0 \cdot \color{blue}{z1}}\right) \]

   4. Applied rewrites 87.8%

      \[\leadsto \color{blue}{z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1} + 0.25 \cdot \frac{\pi}{z0 \cdot z1}\right)} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{\pi}{z1}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z1}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right) \]

      3. lower-PI.f64 51.2%

         \[\leadsto z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1}\right) \]

   7. Applied rewrites 51.2%

      \[\leadsto z0 \cdot \left(-0.5 \cdot \color{blue}{\frac{\pi}{z1}}\right) \]

   if -1100 < z0 < 13.5

   1. Initial program 99.6%

      \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\pi \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) + \pi \cdot \frac{1}{4}}}{z1} \]

      4. add-flip N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \pi}\right)\right)}{z1} \]

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

          \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

      14. metadata-eval 99.6%

          \[\leadsto \frac{\left(z0 \cdot -0.5\right) \cdot \pi - \color{blue}{-0.25} \cdot \pi}{z1} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\left(z0 \cdot -0.5\right) \cdot \pi - -0.25 \cdot \pi}}{z1} \]

   4. Evaluated real constant 99.6%

      \[\leadsto \frac{\left(z0 \cdot -0.5\right) \cdot \pi - \color{blue}{-0.7853981633974483}}{z1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{\color{blue}{\frac{884279719003555}{1125899906842624}}}{z1} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.5%

      \[\leadsto \frac{\color{blue}{0.7853981633974483}}{z1} \]

   if 13.5 < z0

   1. Initial program 99.6%

      \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]

   2. Taylor expanded in z0 around 0

      \[\leadsto \frac{\pi \cdot \color{blue}{\frac{1}{4}}}{z1} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto \frac{\pi \cdot \color{blue}{0.25}}{z1} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto \frac{\color{blue}{\frac{-1}{2} \cdot \left(z0 \cdot \pi\right)}}{z1} \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \color{blue}{\left(z0 \cdot \mathsf{PI}\left(\right)\right)}}{z1} \]

      2. lower-*.f64 N/A

         \[\leadsto \frac{\frac{-1}{2} \cdot \left(z0 \cdot \color{blue}{\mathsf{PI}\left(\right)}\right)}{z1} \]

      3. lower-PI.f64 51.2%

         \[\leadsto \frac{-0.5 \cdot \left(z0 \cdot \pi\right)}{z1} \]

   7. Applied rewrites 51.2%

      \[\leadsto \frac{\color{blue}{-0.5 \cdot \left(z0 \cdot \pi\right)}}{z1} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 97.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1}\right)\\ \mathbf{if}\;z0 \leq -1100:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;z0 \leq 13.5:\\ \;\;\;\;\frac{0.7853981633974483}{z1}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (let* ((t_0 (* z0 (* -0.5 (/ PI z1)))))
  (if (<= z0 -1100.0)
    t_0
    (if (<= z0 13.5) (/ 0.7853981633974483 z1) t_0))))

C

double code(double z0, double z1) {
	double t_0 = z0 * (-0.5 * (((double) M_PI) / z1));
	double tmp;
	if (z0 <= -1100.0) {
		tmp = t_0;
	} else if (z0 <= 13.5) {
		tmp = 0.7853981633974483 / z1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Java

public static double code(double z0, double z1) {
	double t_0 = z0 * (-0.5 * (Math.PI / z1));
	double tmp;
	if (z0 <= -1100.0) {
		tmp = t_0;
	} else if (z0 <= 13.5) {
		tmp = 0.7853981633974483 / z1;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(z0, z1):
	t_0 = z0 * (-0.5 * (math.pi / z1))
	tmp = 0
	if z0 <= -1100.0:
		tmp = t_0
	elif z0 <= 13.5:
		tmp = 0.7853981633974483 / z1
	else:
		tmp = t_0
	return tmp

Julia

function code(z0, z1)
	t_0 = Float64(z0 * Float64(-0.5 * Float64(pi / z1)))
	tmp = 0.0
	if (z0 <= -1100.0)
		tmp = t_0;
	elseif (z0 <= 13.5)
		tmp = Float64(0.7853981633974483 / z1);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(z0, z1)
	t_0 = z0 * (-0.5 * (pi / z1));
	tmp = 0.0;
	if (z0 <= -1100.0)
		tmp = t_0;
	elseif (z0 <= 13.5)
		tmp = 0.7853981633974483 / z1;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[z0_, z1_] := Block[{t$95$0 = N[(z0 * N[(-0.5 * N[(Pi / z1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z0, -1100.0], t$95$0, If[LessEqual[z0, 13.5], N[(0.7853981633974483 / z1), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if z0 < -1100 or 13.5 < z0

   1. Initial program 99.6%

      \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]

   2. Taylor expanded in z0 around inf

      \[\leadsto \color{blue}{z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\pi}{z0 \cdot z1}\right)} \]
   3. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z0 \cdot \color{blue}{\left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right)} \]

      2. lower-+.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \color{blue}{\frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}}\right) \]

      3. lower-*.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \color{blue}{\frac{1}{4}} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right) \]

      4. lower-/.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right) \]

      5. lower-PI.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}\right) \]

      6. lower-*.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \color{blue}{\frac{\mathsf{PI}\left(\right)}{z0 \cdot z1}}\right) \]

      7. lower-/.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z0 \cdot z1}}\right) \]

      8. lower-PI.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\pi}{z1} + \frac{1}{4} \cdot \frac{\pi}{\color{blue}{z0} \cdot z1}\right) \]

      9. lower-*.f64 87.8%

         \[\leadsto z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1} + 0.25 \cdot \frac{\pi}{z0 \cdot \color{blue}{z1}}\right) \]

   4. Applied rewrites 87.8%

      \[\leadsto \color{blue}{z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1} + 0.25 \cdot \frac{\pi}{z0 \cdot z1}\right)} \]

   5. Taylor expanded in z0 around inf

      \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \color{blue}{\frac{\pi}{z1}}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{\color{blue}{z1}}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto z0 \cdot \left(\frac{-1}{2} \cdot \frac{\mathsf{PI}\left(\right)}{z1}\right) \]

      3. lower-PI.f64 51.2%

         \[\leadsto z0 \cdot \left(-0.5 \cdot \frac{\pi}{z1}\right) \]

   7. Applied rewrites 51.2%

      \[\leadsto z0 \cdot \left(-0.5 \cdot \color{blue}{\frac{\pi}{z1}}\right) \]

   if -1100 < z0 < 13.5

   1. Initial program 99.6%

      \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]
   2. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

      2. lift-+.f64 N/A

         \[\leadsto \frac{\pi \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

      3. distribute-lft-in N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) + \pi \cdot \frac{1}{4}}}{z1} \]

      4. add-flip N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

      5. lower--.f64 N/A

         \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

      6. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      7. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      9. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      10. lower-*.f64 N/A

          \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

      11. *-commutative N/A

          \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \pi}\right)\right)}{z1} \]

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

          \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

      13. lower-*.f64 N/A

          \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

      14. metadata-eval 99.6%

          \[\leadsto \frac{\left(z0 \cdot -0.5\right) \cdot \pi - \color{blue}{-0.25} \cdot \pi}{z1} \]

   3. Applied rewrites 99.6%

      \[\leadsto \frac{\color{blue}{\left(z0 \cdot -0.5\right) \cdot \pi - -0.25 \cdot \pi}}{z1} \]

   4. Evaluated real constant 99.6%

      \[\leadsto \frac{\left(z0 \cdot -0.5\right) \cdot \pi - \color{blue}{-0.7853981633974483}}{z1} \]

   5. Taylor expanded in z0 around 0

      \[\leadsto \frac{\color{blue}{\frac{884279719003555}{1125899906842624}}}{z1} \]
   6. Step-by-step derivation

   7. Applied rewrites 50.5%

      \[\leadsto \frac{\color{blue}{0.7853981633974483}}{z1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.5% accurate, 2.1× speedup

Math

\[\frac{0.7853981633974483}{z1} \]

FPCore

(FPCore (z0 z1)
  :precision binary64
  (/ 0.7853981633974483 z1))

C

double code(double z0, double z1) {
	return 0.7853981633974483 / z1;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(z0, z1)
use fmin_fmax_functions
    real(8), intent (in) :: z0
    real(8), intent (in) :: z1
    code = 0.7853981633974483d0 / z1
end function

Java

public static double code(double z0, double z1) {
	return 0.7853981633974483 / z1;
}

Python

def code(z0, z1):
	return 0.7853981633974483 / z1

Julia

function code(z0, z1)
	return Float64(0.7853981633974483 / z1)
end

MATLAB

function tmp = code(z0, z1)
	tmp = 0.7853981633974483 / z1;
end

Wolfram

code[z0_, z1_] := N[(0.7853981633974483 / z1), $MachinePrecision]

Derivation

1. Initial program 99.6%

   \[\frac{\pi \cdot \left(-0.5 \cdot z0 + 0.25\right)}{z1} \]
2. Step-by-step derivation

   1. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

   2. lift-+.f64 N/A

      \[\leadsto \frac{\pi \cdot \color{blue}{\left(\frac{-1}{2} \cdot z0 + \frac{1}{4}\right)}}{z1} \]

   3. distribute-lft-in N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) + \pi \cdot \frac{1}{4}}}{z1} \]

   4. add-flip N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

   5. lower--.f64 N/A

      \[\leadsto \frac{\color{blue}{\pi \cdot \left(\frac{-1}{2} \cdot z0\right) - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}}{z1} \]

   6. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   7. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right) \cdot \pi} - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(\frac{-1}{2} \cdot z0\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   9. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   10. lower-*.f64 N/A

       \[\leadsto \frac{\color{blue}{\left(z0 \cdot \frac{-1}{2}\right)} \cdot \pi - \left(\mathsf{neg}\left(\pi \cdot \frac{1}{4}\right)\right)}{z1} \]

   11. *-commutative N/A

       \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \left(\mathsf{neg}\left(\color{blue}{\frac{1}{4} \cdot \pi}\right)\right)}{z1} \]

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

       \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

   13. lower-*.f64 N/A

       \[\leadsto \frac{\left(z0 \cdot \frac{-1}{2}\right) \cdot \pi - \color{blue}{\left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \pi}}{z1} \]

   14. metadata-eval 99.6%

       \[\leadsto \frac{\left(z0 \cdot -0.5\right) \cdot \pi - \color{blue}{-0.25} \cdot \pi}{z1} \]

3. Applied rewrites 99.6%

   \[\leadsto \frac{\color{blue}{\left(z0 \cdot -0.5\right) \cdot \pi - -0.25 \cdot \pi}}{z1} \]

4. Evaluated real constant 99.6%

   \[\leadsto \frac{\left(z0 \cdot -0.5\right) \cdot \pi - \color{blue}{-0.7853981633974483}}{z1} \]

5. Taylor expanded in z0 around 0

   \[\leadsto \frac{\color{blue}{\frac{884279719003555}{1125899906842624}}}{z1} \]
6. Step-by-step derivation

7. Applied rewrites 50.5%

   \[\leadsto \frac{\color{blue}{0.7853981633974483}}{z1} \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025250 
(FPCore (z0 z1)
  :name "(/ (* PI (+ (* -1/2 z0) 1/4)) z1)"
  :precision binary64
  (/ (* PI (+ (* -0.5 z0) 0.25)) z1))